Have they discussed this with the horses? There are horses who wait to poop and/or pee when they walk into their stall. I remember one that always peed when he was in the crossties. What was disgusting was the instructor. She stood in the middle of the puddle while continuing with her student. Shavings not necessary? I can’t imagine why anyone would expect they can potty train a horse. Is there an aid we should be using? Or maybe two, one for mounted, one on the ground? A lot of horses are keyed up enough at a show to poop while underway. I worked with an instructor I had known for a while who admonished me when we stopped to poop. It is much easier to clean up when it is in a pile. Anyway, I had other things higher up on my list. He didn’t like the part of X-Halt-Salute where he was supposed to stop, for example. One judge’s comment was “nice try.”
One of the reasons you spend hours watching your horse - any horse, actually - is figuing out their foibles. My gelding would spread his hind legs and stretch out, a sign he was going to pee, right? Not quite. He pushed himself up on his tiptoes and then it started. He almost never peed or pooped in the aisleway. If he did I knew he had been inside too long. Usually he would poop and pee when he saw me coming to bring him in, then met me at the gate. If he didn’t do it then, he waited until I turned him out. Poop, pee, then a drink of water, then over to the round bale. He figured all of this out himself. He was on pasture board and the unchallenged alpha for 20 years. The herd would back off the round bale until he found a good spot and then resume eating.
The “culture” in that barn appears to be rooted in lack of training for the instructors, boarders, and other humans. Sounds like they don’t know enough about horses, their behavior, and how they learn. Why else would they expect anyone to successfully train a horse to manage an unpredictable natural activity? Sounds like they haven’t figured it out themselves. If they had, the horse would know it wasn’t okay. Maybe it’s vocal: walk, trot, don’t pee in here, canter, etc.
You can learn so much about horse behavior by standing for 10-15 minutes (longer if you have time). Everyone knows about herd dynamics and pecking order. When my horse was getting old, around 26-27, he started retiring from the alpha role. I was surprised how many people were watching the changes. He delegated things to number two. If a new horse arrived they kept an eye on him. If the newbie started walking, two of them would stay between the newbie and their aging leader.
Watch the horses. Especially how they interact with people.