I don't know if you are aware, but Caliph Abd al-Malik, who built the Dome of the Rock and made additions to Masjid al-Aqsa, initially built the Dome with several purposes in mind.
One of them was to save the Temple Mount from the predations of Christian relic seekers, who joyfully chipped away at the bedrock of the Mount to sell the rocks as relics.
One of them was to build something so beautiful that it would outshine the beauty of the Church of the Sepulchre. While the Church is built with Jerusalem stone and the building is left with the natural colors of the stone, the Dome of the Rock is painted with blues, greens, and black, and the gold dome is an eye-catcher from anywhere in the Old City of Jerusalem.
One of them was to acknowledge that this place was particularly important and holy, but also to display dominance over anyone else who would acknowledge the holiness of the spot.
The Dome of the Rock is not a proper Mosque, except in the strict colloquial sense. It was built for circumnabulation and contemplation, much like the Q'aab in Mecca.
(I did a lot of research on this when I was in college, and this is what I got from Muslim sources.)
If I had to guess... Should the Jews and Muslims in the area come to true peace, it occurs to me that the celebration of the holiness of the area would be the main focus. Therefore, I would also guess that if the Temple was rebuilt, several things would happen.
First: The beauty of the Temple would still outshine the Church of the Sepulchre.
Second: The relic seekers would still be out of commission.
But third, even when the Temple is standing, it would be a place acknowledged for its holiness, and shared - as anyone can bring offerings. Jews man the Temple, but anyone can bring offerings to God there.