My vibes think this could be an XY problem.

The upper section https://www.openstreetmap.org/way/716701652 is already access=private so we already have this info somehow. This is the "X problem".

And yet, if Anon is asking for more access=private coverage, one likely reason is that someone is trying to go up Braemar Hill through here and realize it's private access only, and is therefore complaining. This would then be the "Y problem" that we need to deal with.

However, it is gonna be very difficult to prove the case, since there are several available OSM routers. At least GraphHopper doesn't do that.