First of all, it's a building. Historical maybe but it should be treated with reverence and respect but Our Lord may have been born there, but he isn't there anymore and we are told when he returns he will make all things new. It broke my heart to see what they did. Not so much what was done but their under disregard for the place.
Christians can pray any place. They don't need to go to a shrine to pray. Thhe Muslims hate Christians as much as they hate the jews so this was no surprise to me when it was left like a pig pen. They didn't desacrate the place. They desacrated themselves by their hate that their relgion teaches and stresses. They will answer for it one day and I believe that day is coming soon.
Those of us who maintain the ancient Christian belief in the Eucharistic Presence will vehemently disagree with you there. I hope the Blessed Sacrament was preserved from desecration.