leisure walkers tend not to leave shit allover the path
You say that, but... It is a problem, actually. And a more unsavoury one than a bit of horse shit, too.
This thread is bonkers. I particularly enjoyed the confusion over the differences between a horse (brain the size of a walnut, body of 500kg+) and a dog (larger brain, but maybe 25kg for a medium sized collie type)... Obviously they're completely comparable in terms of control and training. Plus the idea that they should be ridden on farms, fields and woodland - none of which is routinely legally accessible to riders, being, as it is, someone's livelihood... Genuinely hilarious.
I'm all for tolerance and trying to make things work for everyone, but it works both ways. It isn't helpful to focus on banning horses from paths, complain about them on the roads, and insult all horse owners with bizarre stereotypes. There has to be a compromise, reasonable adjustment - segregated access, for example. The collection bags some posters think are the holy grail are not designed for ridden horses - they're designed for carriage horses. They are not designed for horses being ridden at speed (as they might on a track before / after the shared area), and they would affect the biomechanics of the ridden horse in work. They could, conceivably, have a serious effect on the horse's physical well being when used long term (speculation, nobody has done this kind of study). Returning after the fact to pick up droppings in a shared space might be a more reasonable - but imperfect - compromise.
As for the OP, you can't grumble about new housing developments while citing the urban nature of the area as a reason why horses shouldn't be there. If you're rural, or semi-rural, and want the advantages that brings (which the OP seems to appreciate since comments have included "lucky enough" to be on the edge of countryside), you accept that countryside = country ways, animals and shit. At least if you're an urban dweller, who doesn't claim to be "lucky" to be on the edge of the countryside, you're consistent when you complain about horses...