The Lord God of Israel would jus' LOVE to replace the Mosque of Omar with the Third Temple.

But the Jews aren't ready for it:

The secular Jews would set up a profane swine-eatery Macdonalds and bordello on its hallowed grounds.

And the Orthodox religious Jews would quibble forever over who's going to be the High Priest.

Only when a strong Messianic Jewish leader appears will the Mosque go, as per Ezekiel 40-49.

Until then, it's the lesser of two evils that it should remain in Islamic hands, as the Waqf prevent idolatrous Xtian crucifixes and squabbling Jews from profaning it.