There is informal=yes for informal ways https://wiki.openstreetmap.org/wiki/Tag:informal%3Dyes however an informal highway=track seems lees likely than highway=path.

highway=track implies that it's physically usable by vehicles (might only be 4wd, or private maintainance workes, but at least someone could), so if that's the case here then that's fine.

I'd suggest to also tag access modes https://wiki.openstreetmap.org/wiki/Key:access

You said it's not blocked off, so maybe motor_vehicle=yes or permissive?