I don't know about what happened to the church during the great events of the 20th century but apparently no one does, so I'll try to provide some modern context.
It seems likely that like other Greek Orthodox churches this belongs to the Greek orthodox church. I did explore whether there was involvement of government conservation organisation, but I didn't find anything. Private ownership and maintenance seems to be the deal with similar buildings in Israel/Palestine and specifically the Jericho area.
Looking at the legal map and this walking map it seems like the Monastery fits into either Area C or (more likely) the regional council jurisdictional area. This means the area is under Israeli control, whether that be directly or through the settler regional council. So you can assume current Palestinian governance has very little to do with the place.
You can see recent road development done by the Israelis that highlights acknowledgement of the tourist value as well as opposition by the Palestinians towards the road (Israeli presence) and perhaps even the presence of the Greek monks ("Another Palestinian minister said the monks’ presence “gave a deceiving impression” about the status of the land around St. George’s").