Or they don’t want their every move on private property selectively edited to be plastered all over the net?
Or their clients and horse owners do not want their horse or their riding ability trashed for entertainment?
There are two sides to filming others, especially without their permission and/or if they are Minors.
My last trainer, who had many younger clients, had a set policy of no videos of other riders on barn property, without their, or their parents permission.
A no video policy on private property is not necessarily for nefarious reasons. IME and IMO these people don’t try too hard to hide their methods. They depend on observers being unwiilling to speak up, thats what needs to be fixed. Fear of retaliation, being shunned and/or having their reputation or business affected, vandalized…or their horse hurt. Almost like organized crime. It is complicated and exists in every discipline and breed shows.