This trail/path/road is clearly visible on aerial imagery and should remain.  "Private" trails should remain on the map, but access tags can be used instead.  "Private" trails are immensely helpful for fire services and search-and-rescue where knowing some easement is present can mean the difference between life and death.

You can read more (and learn about other reasons) on why deleting "private" paths is not the answer here:
https://wiki.openstreetmap.org/w/index.php?title=Why_we_won%27t_delete_roads_on_private_property