foot=private is not the right tagging to use here, as it means the path is legally not accessible.

The normal tagging for paths on access land is foot=yes unless they are on the council’s definitive map as a public footpath (most aren’t). In that case the tagging should be foot=designated.

There is a precedent for a non-public-footpath on access land being closed for conservation near Broad Crag — see https://www.openstreetmap.org/way/1086252957. In that case we went with disused:highway=path foot=no.

I think that tagging is appropriate here too, so I’ve made the same changes here in https://www.openstreetmap.org/changeset/157705044

https://www.openstreetmap.org/way/1086252957#map=18/54.455882/-3.211548