Relation between a specific POI and an entrance?

Very thoroughly researched! I was aware of the proposed associated_entrance relation (and I've used it myself) but not of the proposed provides_feature relation, which, it appears, is a bit more popular.

I would have suggested associated_entrance, but knowing now that there are two competing proposals -- and best I know, neither has gained enough traction to be implemented in end-user software -- it's hard to know which relation to pick. Certainly doing both seems wrong.

In the situation you describe, I'd be satisfied (for now) with adding the POI nodes near the most useful entrance for each POI, rather than locating them near their interior coordinates.

type=site is less unpopular so to speak. It is used when areas can't describe the feature fully. Although, type=provides_feature author disagree https://wiki.openstreetmap.org/wiki/Talk:Relations/Proposed/Provides_feature without elaboration a decade ago, this has already been used in site=parking widely, with exactly the same usage to relate the =parking_entrance to the =parking.