As a neutral outside observer it’s quite clear to me that the geometry of this path should exist in OSM and be tagged as highway=path + informal=yes. Responsible map renderers should de-emphasize informal paths to distinguish from designated trails (official, sanctioned, etc). Here is an example of AllTrails doing a good job with this. Paths tagged informal=yes are de-emphasized with a much lighter dotted line while paths without this tag get a heavier dashed line. At a slightly lower zoom the informal paths are not visible at all.
This communicates to the map reader that these paths are lesser in some way and may be untrustworthy. The All Trails legend labels the lighter line weight as an “Unmaintained trail” and describes it as “Trails that are not maintained as part of an official trail system and may be overgrown or hard to follow.”
I don’t understand the local laws enough to comment on the appropriate access tag, but I have used access=unknown for situations like this. Unfortunately various OSM validators call out this value as an error. Perhaps they should not.
Landowners being unhappy about a path showing up on a map is not a legitimate reason for the way to be removed from OSM. Instead their frustration should be directed at map style designers using OSM data. Requesting that they de-emphasize or not display paths tagged with informal=yes, access=no, access=private, or access=unknown is perfectly legitimate. Removing the way from OSM is not.