Thanks. Based on https://wiki.openstreetmap.org/wiki/Tag:access=permit

> This tag should be used in cases where a permit is required, but is routinely granted to everyone requesting it. In cases where access to permit is obstructed it would be more appropriate to tag these areas access=private or access=no. So access=permit does not apply where permit is rarely granted, or only granted in exceptional circumstances, or if obtaining it is complex or there is a long waitlist.

It does sound like access=permit is better.