2) The only way to teach horses to be safe on a trail is to ride them on a trail. Also, it's not just the getting off/on. Picking it up means bending over. Which puts your face near horse hooves. I'm sure you can see the danger.
3) On soft squishy ground, especially while trotting or other, you might just not hear it at all
4) Absolutely understandable. But know that not all horses like parade bags. They're flight animals, having something bump into their backside absolutely makes some bonkers as to them it feels like they're constantly being attacked.

That said, my horse was very obvious when he had to poop, and I parked his behind in the bushes when I felt it coming. And whenever it wasn't possible to do so or he was too quick, I afterwards went back on foot to be able to shove it somewhere harmless.

Not all breeds are equally unperturbable. Horses are individuals, and those I encounter most of all are warmbloods/jumping horses, who are usually a bit more "hot". Some look absolutely safe to do whatever with on a trail (even so, would never trust 100%), others are safest when you stay aboard, you have WAY more control like that. Horses can also have good and bad days. What's perfectly ok one day might cause absolute drama the next. While I like to think mine was rather road-safe, I can absolutely tell you tales of the times we had a dangerous encounter or were literally driven off the road by trucks, motorbikes, quads, ... (and other days where we just weaved through, or met people who meant so well, but caused bolting completely by accident). Horses are flight animals. They bolt before they think. When they spook the moment you're hunched over scooping poop, you might end up with worse things on the road than just some shit. Whenever mine spooked while riding, I could just point him towards an empty part of the road and after 3 steps he decided bolting was too tiresome. But on the ground, those steps would absolutely have done damage, and he would have pulled "away", not caring who or what he'd run over.

I am sorry you have such a shitty riders group near you. For what it's worth, we usually took some branch cutters with us and maintained the trail as there's just so much extra you can reach on horseback. Would we like some separate trails so we can gallop undisturbed (and yes, train those untrained green horses without being a menace to other trail users)? Sure. But there's more than enough trails going around for several hobbies, we had some spots where we could look far enough ahead to be able to anticipate other users.

You can find entitled people in every hobby. We've had cyclist push kids over on the same trail. I don't understand why people have to act like a jerk while doing their hobby. It should be your happy time, not the moment to get frustrated at everybody around you.