If the game is meant to entertain the students, and the students are no longer entertained by that particular instance of the game, then why is it so important to the school that students remain inside the stadium?

It really seems to me that the school has somehow made these student/spectators a product in some way. Perhaps it looks better to network television if the stadium stays packed the entire game?

Yes, it obviously is a better image to have your school look engaged, rather than absent, on national TV.

But the other reason (for schools with prominent teams) is that the number of seats available is often less than the number of students who might want to attend, so to be fair to students who actually want to attend, it becomes necessary to disincentive people from requesting tickets if they’re not serious about attending.

It’s pretty common for schools to give out tickets in a lottery system, and then penalize your future ability to successfully get a ticket if you show a pattern of not actually using the tickets. Back when I was in school, this was tracked by scanning the tickets when you entered.