A few points that might help in understanding how horses work.
They are flight animals. In a crisis, they run first and ask questions later. The worst thing you can do is surprise them. If you whiz past them silently with no warning, the first thing they see is you coming close to their side. That is very difficult for them, as all their instincts tell them to get the hell away from whatever it is. A rider can't 'collect' their horse and warn it if they haven't heard you either. The horse will see you first because a horse's eyes can see further behind them than the rider's can. If you can warn horse and rider, not too suddenly, before you overtake, it helps.
Horses can't see the ground for several feet in front of them, their nose is in the way. So, especially if you're on a 'bent, it would be better not to pull in too close in front. If you did, you'd suddenly 'emerge' from under their feet.
Most people wouldn't take a horse on the road unless it was safe in traffic, but horses do have to
learn to cope and there has to be a first time, so remember, like a learner driver, it could be the horse's first time out. Some riders wear tabards saying 'young horse' in such a case.
A horse could have been flurried by a previous bad or aggressive overtake. Some people in cars, vans, or lorries hoot as they pas and rev their engines, or give intentional close shaves, just like they do to us. Just as it upsets us and makes us shaky, it upsets horses, and they don;t understand why it happens. It helps if you give them maximum space.
There are few bridle paths these days that are passable for horses, and getting to them usually involves using roads. If you don't think horses ought to be on roads, then remember the quotation about 'and when they came for me, there was nobody left to speak for me'.
Horses aren't machines. Even the best behaved one on the world can have a bad moment- like deciding not to step on a drain cover or in a puddle where they can't see the bottom, and moving to the right to avoid it. So giving them plenty of space is always good policy -just as you give cyclists space in case
they do something apparently unpredictable (like swing out to avoid a pothole).
No doubt some riders are rude, but please don't condemn the rest for that; we don't like it when we're condemned for a cyclist who behaves badly. Most riders will appreciate you giving them warning and space and passing slowly. If they don't wave a thank you, it may be because taking a hand off the reins isn't the best thing to do when you're trying to keep your horse under close control and reassure it (riders communicate through the reins to the horse via the horse's mouth) and it may not be a good time to shout. Amongst the horsey, a nod of the head or a lift of the stick is the official thank you between riders, and it's easy to miss if you aren't looking for it.
And thanks for reading this far.
PS A rider doesn't know when a horse has left a dropping. They don't stop to do it.