I usually add barrier=* near the intersection of a path and a highway as per the wiki, and add the corresponding access=private/customers/destination tag to the corresponding highways.
I have only used so far access=no on highways that are blocked or are in strongly restricted areas (e.g. military)
I usually don’t add the access=* tag to the barrier node, but I came across an example of a large government estate that has multiple gate access, where there is a main entrance open at specific hours but other gates seem to be closed permanently (or rarely open).
Should I always add access tags to barriers and which one in this specific example?
The Wiki suggests locked=yes for this case (assuming it’s actually locked and not just shut). The rationale is that access= is for legal rights to access and that isn’t the same as physical access. This seems to fit your situation well: you don’t know who is allowed, so you don’t set an access tag, but you can tag that it’s (usually) locked.