It depends on what the on the ground situation is that we are trying to depict.  

I am assuming, based upon your use of access=private, that while the general public is no longer legally allowed to use the trail, there are some people that are, e.g. members of a private country club, a resident of a homeowner’s association, or park rangers. If this is the case, the manner in which the trail is now mapped is correct (with the exception of the bicycle=permission, and foot=permissive tags).   If on the other hand, the trail is closed to everyone but still physically exists, use access=no, leave it connected to the rest of the network, and remove the specific access tags, e.g. bicycle=permissive.  Finally, if the trail no longer physically exists to one degree or another you can use the lifecycle tagging (https://wiki.openstreetmap.org/wiki/Lifecycle_prefix), e.g. disused:highway=path the appropriate access tagging, probably access=no, and leave connected.  In this final case, you could simply delete the trail, but you risk some other mapper adding it back in if it still appears in other sources, such as imagery, the Strava Heatmap, or the USGS 3D Elevation Program data.  There are not many cases where one would disconnect a trail from the larger network of trails and roads.
