Thanks for spotting and updating this from highway=path to highway=bridleway. It was already tagged with designation=public_bridleway, so the access tags really should be horse=designated + bicycle=designated + foot=designated.

Although the iD editor you're using unhelpfully presents all the access tags as if they ought to be completed, access=no is rarely correct or helpful on a bridleway, cycleway, or footway.

There's a bit more information on PRoW tagging here:
https://wiki.openstreetmap.org/wiki/User:Rjw62/PRoW_Tagging