The city recently paved an old railroad right a way. For years we have ridden it and not worried about road apples left on the old gravel. now that's paved and in use by lots more bikers, roller bladers, walkers, mom's pushing strollers etc, I do get off the horse and kick the manure off to the side.
Dirt trails up in the mountain. Even ones frequented by mountain bikers, I don't worry about it. The road apples dry out pretty quickly around here ( Utah is a dry state) and turn into decomposed grass on the trail. I like to think that it helps keep the trails from washing away. But I know the mountain bikers grumble about their tires flipping it up in their faces ( most mountain bikes don't have fenders) But I just rarely every see gooey or juicy road apples. They dry out so fast. And unlike dog crap, Horse manure doesn't not contain pathogens. Thats why Horse and Cow manure is frequently used to fertilize fields and humana nd dog feces are not.
And as most horse owners know. Horse manure doesn't stick to the bottom of your shoe. Unlike dog crap, where I want to throw my shoe away when I step in a pile.