I think these excerpts from a paper, Managing the Divine Jurisdiction: Sacred Space and the Limits of Law on the Temple Mount (1917-1948), by Robert W. Nicholson go far in explaining the roots of what everyone now terms the "historic status quo" that must be preserved there:

...The British chose to govern it “hands off” for fear of upsetting the worldwide Muslim community. The Arab Muslim leaders who controlled it used its autonomous status to create a zone of de facto sovereignty and the center of a would-be independent Arab state in Palestine. The site’s physical impregnability only reinforced its independence. Surrounded on all sides by walls, accessible only by a few gates, and honeycombed with subterranean passageways, the Temple Mount provided a natural sanctuary from government authority...

...The Allied armies breached the perimeter of the crumbling Ottoman Empire and arrived at the gates of Jerusalem in late 1917. The Turks and Germans shortly withdrew northward, and Jerusalem’s Arab mayor surrendered to the Allies on December 9. Two days later, British General Sir Edmund Allenby entered as the city’s first Christian conqueror since the Crusades. Unlike his predecessors, Allenby was careful to demonstrate goodwill to the inhabitants and humility before the hallowed reputation of the city.

Standing on the steps of the Tower of Herod, he announced that Southern Palestine was now under military occupation but that “every sacred building, monument, holy spot, shrine, traditional site, endowment, pious bequest, or customary place of prayer of whatsoever form of the three religions will be maintained and protected according to the existing customs and beliefs of those to whose faith they are sacred.”...

... the British identified the Haram ash-Sharif as particularly sensitive. “The Dome of the rock,” wrote one official, “which was the point of prayer to which Mohammed himself turned, before he established Mecca, gives Jerusalem a special sanctity to Moslems of all sects.” Both before and after Allenby’s entry, policymakers were unanimous that Muslims should be given complete control of the site and all other Islamic holy places. Making this concession, they believed, would soften the impact of the “Crusader victory” and even prove a positive force for Britain’s image. “The effect of the fall of Jerusalem will…be considerable among Moslems as a whole,” one official predicted, “and have a tendency to produce Anglophil [sic] sentiments, and to lessen the malignant power of political Pan-Islamists.”...

...Colonel Ronald Storrs described Britain’s dilemma as “how A should ‘restore’ the property of B to C without deprivation of B.” This dilemma was vividly illustrated at the Temple Mount complex. This sacred space belonged to Palestine’s Muslim community, but it also belonged to the Jews in a way that was difficult to ignore. The space was recognized early as a potential challenge to law and order. Two weeks before the Balfour Declaration went public, Lord Curzon identified the Temple Mount as an insuperable obstacle facing any potential Jewish state in Palestine.

Jerusalem was the only possible Jewish capital, but Curzon knew that Muslims would never allow Jews to take the Haram ash-Sharif. He believed that the only hope for the British was to set up a European-style government, guarantee the safety of the holy places, give control of the Haram to the Muslims, allow Jews to buy land, and protect equal rights for everyone...

...Just prior to Allenby’s entry, a British official in Constantinople suggested that a proclamation be made “announcing that we are the protectors of the Moslem religion and would pay every respect to the Moslem Holy Places.” This suggestion was followed to the letter. “Guards have been placed over the holy places,” Allenby reported to his superiors soon after his arrival. “The Mosque of Omar and the area around it have been placed under Moslem control, and a military cordon of Mohammedan [Indian Muslim] officers and soldiers has been established around the mosque. Orders have been issued that no non-Moslem is to pass within the cordon without permission of the Military Governor and the Moslem in charge.”

Meanwhile, Lord Balfour dispatched a telegram to Sharif Hussein reassuring him that the Allies would appoint a regime “approved of by the world” to manage Palestine’s holy sites. Muslimauthority over Islamic holy sites would be undisputed, he promised. “As regarded the mosque of Omar,” he explained, “it would be considered as a Moslem concern alone and would not be subjected, directly or indirectly, to any non-Moslem authority.”

Britain’s policy of affirmative deference encouraged Muslim autonomy over the holy sites in an effort to show goodwill The government in Palestine pursued affirmative deference under the principle of the status quo. This term of law and art had its roots in the nineteenth century when Ottoman officials had attempted to resolve Christian disputes over their holy sites.

The basic idea was to “freeze” a sacred space in time with all the rights attached thereto. Sultan Abdul Mecid was the first to implement the idea when he published a set of decrees in 1852 on management of the Christian holy sites. The British seized on this attractive “non-policy” and made it the bedrock of their administration. Storrs called it their “strong tower of defence against the encroachments from all quarters.”

From Holy Places in the Israeli-Palestinian Conflict: Confrontation and Co-existence

edited by Marshall J. Breger, Yitzhak Reiter, Leonard Hammer, p. 27

...The Mandate did make specific provision for Muslim sovereignty over Islamic holy places in the country. Article 13 stated, “[N]othing in this mandate shall be construed as conferring upon the Mandatory authority to interfere with the fabric or the management of purely Moslem sacred shrines, the immunities of which are guaranteed.” This provision had stayed intact throughsuccessive drafts of the Mandate, and effectively carved out de jure zones of Muslim autonomy inside Palestine. Two years later, the British enacted the Palestine (Holy Places) Order-in-Council. This law formally removed the holy places from the jurisdiction of Palestine’s civilcourts and assigned resolution of disputes at these places to the executive branch...

...Overall, British policy on the Mount was driven by a fear of triggering religious conflict in the local and international arena. Tasked with managing one of the world’s holiest lands, the British followed a policy of affirmative deference toward the country’s Muslim community...

... Arab desires for self-determination grew stronger [during 1920] as the British regime continued....Muslim leaders inside the Haram prevented the philanthropic Pro-Jerusalem Society from completing a scenic pathway along the ramparts of the Old City by refusing to allow engineers access to parts of the wall inside the Haram. Waqf officials also obstructed efforts to restore the Tariq Bab al-Silsileh, a famous street in Jerusalem. These activities were intended as open assertions of Arab authority over Jerusalem’s physical space...

...the creation [on late 1921] of the Supreme Muslim Council should be seen primarily as an effort to legitimize the British government by incorporating into it the Muslim power structure from Ottoman times. Only a few days after becoming president of the Council, Haj Amin al-Husseini moved his offices from the British government building into the Haram ash-Sharif. This was a major turning point for Palestine’s Muslim community: now, for the first time, a coherent and autonomous Muslim regime held control over Palestine’s most important Muslim space. The Haram gave this regime a territorial center, a source of legitimacy, and a means of mobilizing international Islamic solidarity. It was the only place in Palestine that Arabs could legitimately exclude Christians and Jews...

...The physical restoration of the Haram throughout the 1920s suggests tacit recognition by the Supreme Muslim Council of the site’s legitimizing power. The Haram provided meaning for Muslims in its embodiment of what it meant to be an authentic “Palestinian.” It also empowered communal life by providing an autonomous space where Muslims could conduct their affairs independently of foreign intrusion...

...the Jews did not own the Western Wall. Legally, it was the absolute property of the Muslim community: the Wall itself was part of the Haram, and the alley was part of an ancient waqf dedicated to North African Muslims. Islamic tradition venerated the site as the place where Muhammad had tethered his Buraq before ascending into heaven. Under the Ottomans, Muslim ownership was rigidly enforced. In 1840, government officials had denied a Jewish request to pave the alley since it was waqf property and connected to the Haram. Jews were forbidden to even raise their voices or display their sacred books before the Wall. In late 1911, the trustee of the waqf appealed to the Ottoman government to stop elderly Jews from bringing benches to the Wall. The concern was that it would establish a precedent that later generations might imply as a sign of ownership. Similar disputes occurred in 1912 and 1914. These events show that Muslim attempts to restrict Jewish access to the site had been occurring long before the Balfour Declaration...

And one more concrete example of how the status quo then worked:

On September 28, 1925, Jews brought benches to the Wall for the observance of Yom Kippur. Muslims immediately complained to the government, and Storrs ordered police to remove the benches. The Zionist Commission complained...Storrs...hoped that Jewish and Muslim religious leaders would sort out the dispute for themselves, and tried to convince the waqf to build stone benches in the courtyard to obviate the need for the Jews to bring portable ones. (William Rappard, a Swiss member of the League of Nations Mandates Commission. Rappard suggested to a delegation of Palestinian Jews that “worshippers at the Wailing Wall adopt the method employed by Alpine herdsmen who carry portable stools strapped to their waists.”)...Davar pointed out the perceived legal inconsistencies under the Mandate: “According to the law it is permitted to bring defecating donkeys near the Wall in front of Jews who pray there. But it is forbidden to bring stools…what a sacred law!”...

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After further research, this from the REPORT of the Commission appointed by His Majesty's Government in the United Kingdom of Great Britain and Northern Ireland, with the approval of the Council of the League of Nations, to determine the rights and claims of Moslems and Jews in connection with the Western or Wailing Wall at Jerusalem December, 1930

(2) The application of the principles of Status Quo.

The way in which the Palestine Administration has gone to work in fulfilment of the Mandate for protecting what it has deemed to be "existing rights" in the Holy Places and in other religious buildings and sites, has been by seeking to maintain the status quo, of which the principles and machinery will be briefly explained below.

As regards the Holy Places, in the restricted sense of this expression, the Administration both before and after the setting up of the Mandate have applied the same rules of status quo as were in force before the War, i.e., the rules based on the firman of 1852, which in its turn is for the most part a mere confirmation of the status quo of 1757. As apportioned between the three principal Christian Rites, viz., the Orthodox Greek Rite, the Latin (or Roman Catholic) Rite, and the Armenian Orthodox Rite...The same principles for conserving the status quo have been applied by the Palestine Administration with regard to the Western Wall. Here too the Administration has had in view the maintenance of the status quo ante bellum, as far as it has been possible to ascertain what that consists in.

In the White Paper of November, 1928, the British Government stated to Parliament with great clearness what principles they consider to be the leading ones to be followed in treating the points in dispute between Arabs and Jews. From the said document the following paragraphs especially must be quoted in this connection:

The Western or Wailing Wall formed part of the western exterior of the ancient Jewish Temple; as such it is holy to the Jewish community and their custom of praying there extends back to the Middle Ages and possibly further. The Wall is also part of the Haram-al-Sharif; as such it is holy to Moslems. Moreover, it is legally the absolute property of the Moslem community and the strip of pavement facing it is Waqf property, as is shown by documents preserved by the Guardian of the Waqf. The Jewish community have established an undoubted right of access to the pavement for the purposes of their devotions but, whenever protests were made by the Moslem authorities, the Turkish authorities repeatedly ruled that they would not permit such departures frown the existing practice as the bringing of chairs and benches to the pavement. It is understood that a ruling prohibiting the bringing of screens to the pavement was given in 1912.

"The Palestine Government and His Majesty's Government, having in mind the terms of Article 13 of the Mandate for Palestine, have taken the view that the matter is one in which they are bound to maintain the status quo, which they have regarded as being, in general terms, that the Jewish community have a right of access to the pavement for the purposes of their devotions, but may bring to the Wall only those appurtenances of worship which were permitted under the Turkish regime. Whenever the Moslem authorities have preferred complaints that innovations have been made in the established practice, and the Palestine Government on enquiry have satisfied themselves that the complaints were well-founded, they have felt it their duty to insist that the departures from practice which gave rise to the complaints should be discontinued."
Accordingly the British Government has held that the Western or Wailing Wall is sacred to both Parties, and than even though one of them has the exclusive legal ownership of the Wall, yet the other during the Turkish regime and in previous years before the Great War enjoyed the right of free access to the place as to a religious site.

The British Government and the Palestine Administration have apparently, when acting on the said principles, been anxious to maintain the status quo ante bellum in the relations between the two Rites which both have a religious interest in the same spot. From this point of view the supervision exercised by the Palestine Administration in their task of guarding the status quo has been carried out in two directions: on the one hand they have sought to check the Jews from bringing to the Wall appurtenances that are contrary to accepted usage, and on the other they have tried to exclude innovations on the part of the Moslems that may result in a hindrance to or cause disturbances in the carrying on of the customary devotions of the Jews at the Wall.

The Palestine Administration had to take action for the said purpose on special occasions in 1925, in 1928, and in 1929, which occasions are detailed in the Shaw Commission Report.

As stated in the above-mentioned White Paper of 1928 action of that nature has had to be taken immediately in order not to give rise to any infraction of the status quo. In respect to the Jews the prohibition has been enforced against the bringing to the Wall of any benches, chairs, or stools, carpets or mattings, or any screens or curtains for the purpose of separating men and women. On the other hand, in the Rules promulgated in 1929, the Jews are given permission temporarily to bring to the Wall certain appurtenances of worship, duly specified in detail.

and then we hear an exact echo of today's situation:

As regards the terms of the Mandate it is true that in Articles 13, 15 and 16 the principle of religious liberty is proclaimed and that Article 13 especially provides for "free exercise of worship" for all concerned. But from this general rule the conclusion cannot reasonably be drawn that the partisans of any special confession should have the right to exercise their worship in all places without any consideration to the rights of others. If that were so then the whole structure of the status quo in the Holy Places and other religious sites would break down. In the present case the difficulties are aggravated by the fact that the religious site is itself a Moslem Waqf enclosed in and surrounded by other Moslem Waqfs, of which one contains a shrine of the greatest sacredness to all Moslems.

If the Western Wall and the Pavement in front of it ought to be protected in the religious interest of the Jews, due consideration ought also to be paid by the Jews to their hosts, the Moslems, whose sacred shrines have been guaranteed immunity by the terms of the Mandate.

Hence the Commission concludes that the established custom should be a proper basis for deciding the existing rights of the Jews at the Wall. From this it does not follow that the Commission must go back to the primitive forms which characterized the prayers and the conditions at the Wall at the earliest stages. On the other hand the Commission thinks that usage, in order to serve as a basis for a real right, must be of fairly long standing.

and this, too:

the [Mifti-headed Mulim] Council concluded their letter by saying: "The aim of the letter dated 20th July, 1926, was that the necessary steps be taken to prevent the Jews from putting anything in the Buraq, especially on Saturdays and Jewish feast days." On 25th August, 1926, the District Officer wrote to the President of the Council in reply to the above letter as follows: "That the measures referred to in the last paragraph of your quoted letter have been taken, and that no change in the status quo will take place."

... the Commission has come to the conclusion that, although there have been different opinions as to what was allowable under the status quo, both parties in the disputes that led up to the White Paper of November, 1928, based their arguments on the acceptance of the principle of status quo as relevant for their existing rights.