And what is the default fallback for those values that they do not account for ? Access=no or yes ?
Adding subtags is quite common (e.g. tourism=information, information=board; board_type =xxx, we do not tag that as tourism=information_board_xxx)

Here, the basic message we try to convey with the access tag is that you need permission in some form to access the premisses. So access=private (as opposed to access=public). If you want to specify the private group further, that can go in a subtag without breaking any data consumer, and making it easy to add new groups (management, car-poolers, employees, visitors, etc…).
This is the same as with the information board.

Similar amenity=shelter + shelter_type, historic=memorial + memorial:type/memorial ,etc. etc.