Monks can be rowdy folk, especially Eastern monks it seems, meaning no disrespect.
Strictly speaking of course the most appropriate guardian of these holy sites would be either one of the Eastern Catholic Churches or else, in light of crusade-era history, the Latin Rite Church and in particular the Franciscans. But of course non-Catholic Churches don’t agree, and so there is tension and scandal.
Obviously the current arrangement to try to share the holy sites is very imperfect, but it would be very difficult to negotiate a new arrangement. Hopefully these very scandals will serve to encourage all parties to be more prudent, or even charitable if they can manage it.
By the way, the current arrangements for sharing the buildings are often very old, complicated, convoluted things which could give more meaning to an act than appears on the surface. For example, I don’t know but it would not surprise me if by not preventing a monk from one Church from washing the steps traditionally held by another Church the other Church would be ceding control of the steps to the one doing the washing according the mutually agreed-upon rules, and so the scuffle would really have been a minor little crusade of one Church against another, trying to gain control of one tiny piece of the holy land.