Am I Muslim? No. I'm a Jew & a Zionist, though assuredly a liberal one. I believe Israel shot itself in the foot settling in the West Bank (& Gaza before). It created a poison pill of territory that will ensure conflict for decades to come.
As for the Al Asqa Mosque, I'm just pointing that a) it was typical of Islam to construct mosques on sacred sites they conquered. There was no especial bias in using the Temple Mount, which at the time wasn't serving in any formal religious function. And b), as far as I know, Jewish law would forbid construction of a temple or synagogue on grounds used for idolatry, but Islam isn't idolatrous. Obviously experts in Halacha would differ on this point.
The issue of who was here first or to whom the ground is more sacred is meaningless. Ze'ev Jabotinsky, the founder of Irgun & spiritual progenitor of Likud, acknowledged in his book, The Iron Wall, that Jews were taking land from Palestinian Arabs that was deeply sacred to them. He made no pretense (as many on the modern right & even left do) that Palestinians only cared for the land after it fell into Jewish hands. The history of Zionism is far more nuanced than most realize.
Getting back to Ilan Omar, I'm not aware that she hates Israel, but in her capacity as a lawmaker in a body that provides Israel substantial support she's well within her rights to critique its policies regarding a people who are essentially in subject status to it. This tendency to dismiss dissenting opinion because of one's opponents' religion is misleading & rather silly.
Oh, one final point here: You seem to imply that Islamic conquest of Jerusalem by force of arms somehow invalidates any moral right to regard the mosque & grounds as sacred, yet how did Jerusalem become a Jewish city in the first place? How was returned to Jewish control?