You don't really need to add the note or bicycle and horse tags, as a public footpath correctly tagged as highway=footway + designation=public_footpath is already pedestrian only for routing purposes. If people are riding horses or bicycles across the golf course, it is unlikely to be because OSM-based routing software sent them that way.

If it had been tagged as highway=path, then you might have needed horse=no and bicycle=no

https://wiki.openstreetmap.org/wiki/OSM_tags_for_routing/Access_restrictions#United_Kingdom