One thing I will say ... I have always had a foot in both camps - both sorts of riding.
Cyclists have had a right to ride on 'our' bridleways for decades now, and most of them are appreciative of that and considerate of the original users.
BUT
and it's a big but ...
horses are NOT permitted on bike paths or so-called multiuser paths, and in many areas bridleways are being, or have been, substantially altered (eg tarmaced, concreted or have compacted stone chippings laid) such that they are made increasingly less suitable and more unwelcoming - even damaging! - to the horse ... In other areas bridleways are so heavily - and often irresponsibly - used by mtbs (and illegally by trail bikes) that they become dangerous for horses, more 'moderate' cyclists and walkers. I really don't care very much about the walkers - they have over 80% of the PROWs exclusively for their use and and their use alone, and can easily find another route - but I do care about the horse riders and the more responsible cyclists who don't want a path to be ruined for its legitimate users.
So resentment does happen in some areas; hardly surprising.
All that said, though, I have had far more 'run ins' with obstreperous 'ramblers' on bridleways, especially up in the hills, than I ever have with any type of cyclist. and on several occasions I have had to intervene when plain nasty walkers tried to threaten harmless kids merely having fun on their mtb on a bridleway where they had every right to be, and doing no harm to anyone. But intervening with a group of red-socked nasties trying to bully a youngster, from the position of being a mature adult mounted on a well-trained horse, is really quite easy and a great boost to the ego!