"Do you think Muslims care a whit about it?"
Two points:
1. Not all Palestinians are Muslim. Many are Christian, and have probably worship in the Church of the Nativity. (I don't know about the gunmen obviously.)
2. Many Muslims have a strong devotion to Mary and to the events of the Nativity. The site, without the presence of the church that is already there, would probably have become a Mosque - and as a result of their reverence for Mary and the Birth of Christ, not out of a desire to drive out the Christians, though such action would have such an affect.
We are the hostages, and the Holy Places themselves. Israel knows this and proceeds accordingly.