Some time ago I read an explanation of the "crozier on the door" from the USCCB. It has nothing intrinsic to do with the rite of installation. It has its origins in medieval times of a Bishop "taking possession" of a new church and resumed in recent times. Though not in the proffered explanation, it exemplifies the power of the hierarchy, selected outside the community, to bleed a community of its wealth, its labour, and thank him (and God) for it.