No, it's clearly designated as a public bridleway and was tagged as such. What you would like it to be is immaterial.

Please familiarise yourself with access rights on public rights of way in England and how they are tagged in OpenStreetMap before editing any more PRoWs.
https://wiki.openstreetmap.org/wiki/User:Rjw62/PRoW_Tagging

I have reverted your edit and changed it from highway=path -> bridleway