being an assistant high school coach, I would definitely leave in the case that my head coach was a jackass that couldn't be professional enough to have an adult conversation with only the other party around. Nobody likes being embarrassed, especially when the young ******* who screws up himself doesn't get any type of criticism in return because their is no one above him. It makes the entire daily routine dreadful for all participants. Somebody else mentioned that you wouldn't like it if your boss came in and screamed at your mistakes in front of your co-workers; this is totally different because the co-workers you mentioned are kids that you are supposed to lead. 18, 19, 20 year olds that are either going to hate the Head Coach and take up for you, or take the Head Coaches side and think you're an idiot which probably doesn't help the team at all. But hey, that's just my opinion.