Not saying this was a good idea by any means, but I would have imagined that Crusaders, having ridden all the way from Europe to free Jerusalem from Muslim rule, would take the first opportunity to remove all traces of that rule, starting with the Dome of the Rock.

Likewise for the Muslim conquerors and the Christian sites.. Though perhaps their view of Jesus as a prophet and Abraham being a common father son with the Jews prevented that. I did hear that Omar refused to step inside the Church of the Holy Sepulchre, saying future generations would take it as an invite to demolish the church, do he prayed outside and a mosque note stands where he prayed.

Again demolition would have been terrible, just curious as to the mindset at the time.