The unacknowledged problem is that the quality of coaches who give private lessons varies just as, if not more, widely.
The big problem is that every private coach thinks they know something, and quite a few don't. A lot of school coaches who don't know a lot also aren't trying to change too much, so they won't do a whole lot of damage.
Yes but generally a private coach who is an idiot fazes out because they aren't helping players. But a school coach who is walking and breathing can likely get a job for a long time if they want it.
But you're right plenty of private coaches who over-estimate their knowledge.