Good points!
I think the page should be updated on multiple places such that there is a clear distinction is made between access-tags in combination with roads/paths (Key:highway) for which applies:
Access values describe legal permissions/restrictions and should follow ground truth; e.g., signage or legal ruling and not introduce guesswork. It does not describe common or typical use, even if signage is generally ignored.
and things like barriers, ford=yes, railway=crossing, entrance=yes for which the primary criterium is physical access (typical use), although in the exception there is a sign, something like motor_vehichle=permit also can be used
The third category of usage is in keys like information and amenity where these “access tags” are used to indicate the usage is designated for the given “transport mode”
I added something like this also to the Talk page, Three subtypes of usage
Update: I did update: