An Israeli woman places her hands on the Western Wall as she prays at Judaism most sacred site in Jerusalem's Old City.
The Western Wall is not "Judaism's most sacred site." It is a small part of the retaining wall for a platform that encompassed Judaism's holiest site, the Temple Mount and the site of the "Kodesh K'dashim" the "Holy of Holies", that is within.
It is easier for many to pretend that the Wall is the holiest site, because then they can guess that it is a trivial solution to divide Jerusalem and give the Temple Mount to the Muslims and the Wall to the Jews. When they admit that the Temple Mount is holier to Jews, then people might start to wonder why the Dome of the Rock was purposefully built on top of Judaism's holiest site. They might start asking questions about whether the Koran mentions Jerusalem, or whether Mohammed ever flew there on a winged horse. They might start to see analogies between the Islamic destruction of Buddha statues, Hindu temples and Jewish holy sites.
They might even start to wonder why the Dome of the Rock, if it was such a holy Muslim site, was pretty much in ruins before the 1920s.
No, it is less messy to keep pretending that the Wall is Judaism's most sacred site, and that the Jews have no religious claim on the Temple Mount.