Horse riders should have to clean up after their horses on trails.

I'm a big believer in leave no trace in the outdoors. To the best of your ability, everything should be as you found it when you leave the woods.

Wild horses have been extinct in north America for many thousands of years, in my local area as far as fossil and archeological records can show any native horses that might have existed here were long gone before the first native Americans set foot here. They are not a part of the ecosystem.

I don't care if it's biodegradable, I wouldn't leave apple cores and banana peels behind either.

The environment in my local parks isn't so delicate that a few entitled rich assholes leaving behind horse shit probably isn't going to make a significant impact, but there are other places where it absolutely could, throwing off the chemical composition of the soil, contaminating ground water, causing algal blooms, introducing non-native parasites, bacteria, and pathogens, etc. and you should be following best practices across the board. Treat every inch of the outdoors as if it were potentially vulnerable and don't try to bend the rules just because you think you can get away with it.

And it's just an eyesore and detracts from the natural beauty.

The horse people fire back about how they can't carry a shovel with them, or how they may not be able to safely get on or off the horse. This is the shit horses were bred for- to carry people and stuff, I can find you an avalanche shovel and a small folding step stools that will break down plenty small and light enough to fit in a backpack or lash to the saddle with some rope to pull the stool up after you get on, and it's all gonna weigh a lot less than the armor, and rifles, bedrooms, etc. that people used to ride horses with for a lot longer and harder than the couple house you're spending plodding along the trail. If you can afford to go horseback riding you can afford the hundred bucks or so for a shovel and a step stool. If that's not enough for you to get on and off your horse safely on the trail, maybe you should take that as a sign that you shouldn't be riding a horse there, stick to a dude ranch where some big strong cowboy will help you get on and off of the horsey.