I have yelled at students, of course.
But it’s been when they are doing some dangerous (getting ready to jump an oxer backwards was one) or being unkind to a horse. (That’ll get me yelling pretty quick.)
My wonderful, late mentor never yelled. My favorite technique of hers was that she would call a student into the middle of the ring for a quiet chat about why they weren’t following her instructions. Usually it was a misunderstanding that they clarified out of other’s hearing. Occasionally it was laziness or willfulness, in which case, she would give them the option of dismounting, explaining that she couldn’t have people who wouldn’t follow instructions ride and ruin her school horses. But it was all done calmly and quietly, without the side of public humiliation.
I think this is more common than we’re willing to admit. Lots of top riders have been riding at a high level for so long, and have developed such feel, that they don’t consciously understand what they’re doing to influence the horse, they just do it. And don’t understand why others can’t without detailed instruction. As a riding student, I have to understand the mechanics and the how of everything before I can do it. So I’m pretty good at breaking things down for my students as well.
In general, I think we tend to conflate good rider with good teacher, and it’s just not necessarily so.