“The Temple Mount remains, under the terms of the 1994 Israel–Jordan peace treaty, under Jordanian custodianship”
https://en.wikipedia.org/wiki/Temple_Mount
From the Wiki article that you linked to:
“The site remains within the area controlled by the State of Israel, with administration of the site remaining in the hands of the Jerusalem Islamic Waqf.”
Israel has sovereignty over the entire area, and isn’t giving it up - nor should it.
Peace between the Paleoswinians and Israelis will never occur, no matter how much Israel gives up, so long as the Paleoswinians utterly refuse to recognize Israel’s right to exist. The Paleoswinians STILL favor a one-state solution - THEIR state. That is just a non-starter, but they keep teaching this to their children, perpetuating the seemingly eternal state of war. Frankly, giving up land, or even appearing willing to give up land, is a major sign of weakness in the eyes of the Paleoswinians. Israel has, hopefully, learned this lesson and will refuse to give up any additional land.