The caliphates themselves weren't interested in disasembling the holy sites.

Caliphates themselves don't have an explicit social policy but rather they have their fiqh which is then judicially applied by the state. And so it could do most anything to other holy sites depending on the circumstance and circumstance did bring Al Hakim about. And so the caliphates being there enabled it to be possible, which leads back to what I said: "without those caliphates there wouldn't be an attempt to disassemble holy sites in Jerusalem and so no crusaders."

The crusades were started because Alexios wanted to regain his lost territory.

Actually securing Jerusalem for the rebuilding and protection of the Church of the Holy Sepulchre was the main goal. Constantinople was a secondary goal. Hence most forces leaving after Ascalon and not giving lands back to Constantinople.

I definitely do know Alexios spun lies of cruelty from Muslims. I know the Muslims were aggressively expansionist but still that was uncalled for.