Wasn’t it about 6 years ago that the “Religion of Peace” members in Nazareth tried to take over the land around the Church of the Annunciation? They seemed to think they could take over the shrine entirely.
As for the Church of the Nativity, the worst event to the shrine over the past decade was when radical Muslim terrorists took over the church, and while occupying it desecrated it.
To build a giant mosque next door if I recall.