It turns out that most Muslims never believed that until the 1929 riots and after Israel built the plaza in front of the Kotel in 1967.
The Shaw Commission report of 1930, about the riots that started at the Wall in 1929, describes the competing religious claims to the wall - and does not mention the legend of the Buraq at all, only that it is part of the Haram esh Sharif like all the other walls.
One of the Holy Places in connection with which it has not infrequently been necessary to give rulings of the character indicated above is the Western or Wailing Wall; in Jerusalem. This Wall; forms part of the western exterior of the ancient Jewish Temple; being the last remaining vestige of that sacred place it is regarded with the greatest reverence by religious Jews, whose custom of praying there extended back to at least the Middle Ages. ....The Wall is also part of the Haram-esh-Sherif, which is an Islamic place of great sanctity, being reckoned next to the sacred cities of Mecca and Medina as an object of veneration to Moslems.
If the Kotel was considered the holy "Buraq Wall" by Muslims at the time, certainly that would have been mentioned here. Although it must be mentioned that the testimony included the idea that the Muslims had exaggerated the importance of the Buraq Wall:
Merriman introduced part of Chancellor’s statement to the Permanent Mandates Commission in July, in which he stated that the Moslems are trying to invest El Burak with a sanctity never before attached to it. Merriman implied that the Government was as skeptical as the Jews regarding the sacredness of the pavement to the Moslems.
Wikipedia points to an obscure German source from the 1860s that appears to say that the Muslims claimed the entire Western wall of the Temple Mount to be the Buraq Wall.
A 1932 academic paper in the Journal of Biblical Literature, by C. D. Matthews, discusses the topic of where Muslims believe Mohammed tied the steed and entered the Temple Mount:
Among the orthodox who maintain the legends of Islam as historic truth, the famous "Night-Journey" of the Prophet sanctifies, as could nothing else, Jerusalem as a city of Muslim shrines, especially the Rock itself, from which Mohammed ascended into Heaven, and the Wall of al-Buraq, where Gabriel, the angelic conductor, tied up the divine steed on the arrival at the "further mosque" from Mecca. Unfortunately, this last spot is identified as the same as the "Wailing Wall," involving a clash of religious sentiments which has often led to tragic results. Of course, the factious can always find factors. But is the Wall of al-Buraq the same as the Wailing Wall? The spot of contention is the southern end of the western wall of the Haram area. Muslim tradition is now well established that this is the holy station of their Prophet. Some accounts of the Night-Journey, as for instance our own author, say that Mohammed entered through a "gate through which the sun and the moon incline"--or shine at setting. This would indicate, of course, a western or southwestern gate. It agrees with the ordinary identification as the now walled-up Bab an-Nabi underneath the Gate of the Moors, Bab al-Maghribah, just south of the Place of Wailing, in the western wall of the Haram.
He's saying that the Muslim consensus in Jerusalem was that the Buraq was tethered to the south of the Kotel, under the Mughrabi Gate - not where the Kotel is, north of the gate.
But that's just the beginning:
Further, Ibn al-Fakih (903) says the place of the tying up of al-Buraq is in the angle of the southern minaret-which was at the southwest corner of the Haram. And Ibn 'Abd Rabbih (913) says it is under the corner of the masjid-which Le Strange (in his book Palestine Under the Moslems) takes to mean the Aqsa but which can as well refer to the entire Haram and therefore Aqsa, mean the same as the statement of Ibn al-Fakih.
But Muqaddasi (985), a citizen of Jerusalem and a most careful writer, speaks of the "two gates" of the Prophet, in such terms as positively to identify his choice as the ancient double gate in the south wall of the Aqsa. There used to run here a large entrance which is still a subterranean opening from within the mosque. The location of the former double gate is just as near the southern minaret on the southwestern corner as is the Gate of the Moors in the southern portion of the western wall.Further, Nasir I Khusrau (1047), despite the earthquake which came between Muqaddasi and him, resulting in changes of structure and name, still speaks of this gate under the Aqsa as the gate of the Prophet. He says (as quoted by Le Strange, P. M., pp. 178-9): "The gate of the Prophet... which opens toward the kiblah point-- that is, towards the south... The Prophet... on the night of his ascent into heaven, passed into the noble sanctuary through this passageway, for the gateway opens on the road from Makkah." It is only when we come to Mujir ad-Din, as late as 1496, an author whose work was almost entirely of secondary material, that we have a definite change of reference to the southwestern gate in the western wall as the gate of the prophet; and even here the author is speaking mainly of the Gate of the Moors over the walled-up gate which he says incidentally is also called Bab an-Nabi. (See Le S., p. 182.)Finally, Le Strange (p. 182), who had studied thoroughly all the Arab geographers and historians on Palestine, takes the gates of the Prophet as named (not in order) by the two earliest writers Ibn al-Fakih and Ibn 'Abd Rabbih, to be identifiable with those named in the southern wall by Muqaddasi and Nasir. This places the weight of testimony on the western portion of the southern wall of the Haram area, not the southern portion of the western wall, as the proper Wall of al-Buraq of Muslim tradition.
The blue section is where the Kotel is, the green is where the Buraq Wall was accepted as Muslim tradition as of 1932, possibly at the southern corner, and the orange outline covers areas where the original Muslim traditions placed the Buraq Wall.
So when Palestinians today claim that the Western Wall is really a holy Muslim Buraq Wall, they are lying. And it is a lie about Muslim tradition itself, meaning that they are willing to change their own history and legends just to take away any Jewish claim in Jerusalem.
