Yeah, as many others have pointed out this analysis is facile verging on just wrong.
Do universities not compete for students? Why have costs gone up so much faster than inflation? It's not just a question of there being a huge pool of money, you can buy a car with loans and car prices haven't skyrocketed.
The answer is that education is a weird "good" that's probably a Veblen good, among a lot of other reasons.