access=no would imply it's unusable by any vehicle including fire trucks, emergency services, and NPWS maintenance vehicles. Is that the case? (this would be motor_vehicle=no) Or is it just closed to general public vehicles? (this would be motor_vehicle=private).

Additionally do you know the situation for bicycles/mountain bikes? Because a top level access=no also captures bicycles. If the signage says something no "no access, walkers only" then access=no + foot=designated would be okay, if the sign just says no vehicles or maintenance vehicles only then motor_vehicle=private + foot=yes + bicycle=yes might be better.

There's some documentation at https://wiki.openstreetmap.org/wiki/Australian_Tagging_Guidelines/Cycling_and_Foot_Paths#Vehicle_Access