No Obligation-
My point is they absolutely can quit the team and do their own thing. However, they should not then complain they can't run the championship high school season which has a pre-requisite of being on the team.
If you quit said job for said crappy boss you do not continue to get health insurance or paid vacation with said company.
To answer your question in full their are certainly times when you should continue working for a crappy boss and times when you should not. What makes the boss crappy? Attitude, knowledge, resources, your salary, your opportunity for advancement...etc??
Sometimes we perceive things as being "crappy" because we don't know how to handle criticism or getting yelled at.
In this particular situation I have an issue with this teenager's behavior involving them trying to dictate to their coach their terms to remain on the team. Perhaps if they had approached the situation less confrontationally and more in the vein of wanting to work on the situation together or some other alternative (perhaps they did, it was not explicitly said one way or the other) then I might understand more.
However, the bottom line is that the coach was hired by the school to coach the team. If the student does not want to be coached by the school appointed coach that is their prerogative. Consequently they are voluntarily choosing to quit the team and not be a part of the high school TEAM championship and regular seasons.