Isn’t it basically PR to attract students, who in turn will pay very expensive tuition (often backed by student loans, and guaranteed by the government to boot, meaning it can be cranked up really high)?

That’s a great point. It could be seen that way. Though it’s certainly not how it’s usually explained, and I don’t think most people would see that as a satisfying answer. “Sports are profitable because they allow us to charge you more” just doesn’t have the same ring.