I  agree that suitability for bicycles is not a deciding factor for the suitability of of highway=bridleway. But this isn't the question; the question is whether a multi-user path intended for pedestrians / cycles / horses / wheelchairs is best tagged as a bridleway.

The accusation of tagging for the renderer is unfair - this implies using incorrect tagging to get a specific effect. The original tagging (not mine) was not incorrect. CyclOSM has an easy job, since it ignores the horse access tags. The interpretation of tags by Carto is more relevant since it needs to consider them all.
Here `highway=path` + `bicycle=designated` is rendered as `highway=cycleway` THEN (as a lower priority) `highway=path` + `horse=designated` is rendered as a bridleway, whether a way is intended for bicycles is a key factor.

Anyway, we are clearly not going to agree, so I'll open an issue on the UK Community Discussion Board.
https://community.openstreetmap.org/c/communities/uk/86
 