Welcome to OpenStreetMap.

Unfortunately, deleting the highway=path tags here leaves data consumers with access permissions attached to nothing. You'll notice that they've completely disappeared from the map following your edit(s).

If it's a permissive footpath and isn't used by estate vehicles, I suggest using:

highway=footway 
foot=permissive
designation=permissive_footpath
(no other access tags needed)

If it's used by estate vehicles, you might use:
highway=track
vehicle=private
foot=permissive
designation=permissive_footpath

You could also add the following to indicate who's responsible for the permissive nature of the path:
operator=National Trust
operator:wikidata=Q333515

Please let me know if you need any help reverting or repairing these edits.