Watching that video .... seems like much ado about nothing to me. I did not see a lowered shoulder. More like a flying elbow that mostly missed the target.
There are coaches that act like a$$ clowns on the sidelines, and they get away with it. There are a lot of coaches that are less than honorable in the way they manage their teams during practice, recruiting, etc., and they get away with it. They do so because they are not breaking a rule (or not doing so in a major way), and because they don't embarrass the institution that they are representing.
I don't think Princeton had a lot of options in this situation. As you noted, the elbow that Coach Bates threw at Brown University's John Yozzo-Scaperrotta wasn't a great shot. However, it was the head coach of Princeton striking a player of an opposing team. Princeton couldn't act as though that were acceptable, and allow Bates to continue as head coach. They could have suspended him for some interval. Or, they could have fired him. They probably worked through the pros and cons of each in their discussions, and in the end felt they could not allow him to continue as the head coach.
From the NCAA rule book:
Rule 2 Game Personnel
Coaches
Section 4
The head coach shall act in a courteous manner and also shall endeavor in every way to achieve the same from team members. The head coach also shall be in control of and be responsible for the actions of any and all people officially connected with his or her institution.
Coaches shall assist the officials in keeping the game under control. It shall be their duty, upon the request of an official, to control effectively actions of spectators not in conformity with standards of proper conduct.
Rule 5 Personal and Expulsion Fouls
Expulsion Fouls
Fighting and Flagrant Misconduct
Section 14
Fighting is defined as a player, substitute, nonplaying member of a squad, coach or anyone officially connected with a team deliberately striking or attempting to strike anyone in a malicious manner, or leaving the bench or coaches area during an altercation.
(I won't copy all the rules on the penalty, but basically you are booted from that game, can not be on the bench, on the sideline, or with the spectators, and you have to sit the following game as well.)
Rule 6 Technical Fouls
Conduct Foul
Section 7
A coach shall not enter the field of play without the permission of an official, except to attend to an injured player or to warm up a goalkeeper, or during halftime.
So, the head coach is responsible for the behavior of all players, support personnel, and fans of his institution. In addition, he's not allowed onto the field without the permission of the ref. And, it's a violation to deliberately strike anyone (even someone on your team). Coach Bates didn't just push things a bit, he violated two rules, and showed that he could not be responsible for his own actions, let alone his teams, and as such violated a third rule.
If video of the incident didn't exist, and a ref had not noticed it (which is quite possible, as it was away from the field of play, and where the ball was), it's quite possible that nothing would have happened to Bates. With the action being captured on video, Princeton only had a couple of options, and I think they went with the best option.
