The NCAA may be a private organization, but the constituent universities receive money from the federal government in the form of guarantees of student loans and research grants. You may figure, “meh, who cares... go without the government support”, and some universities heroically do this, but try supporting a student body large enough to sponsor NCAA sports while telling ALL of your graduate students and professors that if the government is who would sponsor research in your topic of inquiry, forget it, you can’t do the research because we have no funds. Alumni support scholarships and tuition, but not research.
You left out state taxpayer support!
Most NCAA schools have athletic departments that lose money. The relative few that don't are in the black solely because football revenues support the other sports. In other words, there isn't this huge pot of money that NCAA schools have from being "in the black" on sports.
If you removed/eliminated the loans that students take out for their education, you would just be pulling even money out of those schools. So there would be even less net money to pay athletes. In other words, the claim that the federal government somehow created the NCAA system of not paying athletes is simply ridiculous.
The thing that bugs me about this issue is that so many so-called conservatives appear perfectly willing to adopt Marx's Labor Theory of Value. That somehow, because these students are the ones actually playing the games, that they are the ones who should be getting the revenue.
But that ignores everything else that goes into putting on a competitive athletic program. Just for starters, if you separated every college football program from its sponsoring University and just have them enter a private league, the revenues would absolutely collapse. The alumni and students that are the core of a football program's support only exist because of the University structure in the first place. Eliminate the team's link to the university, and you've torpedoed their market. You'd also have eliminated alumni cash support for the programs.
Then there is the issue of having enough players to feel the team in the first place. Even on the very highest level division 1 programs, the majority of players are not drafted into the NFL. Yet, their participation in the program is essential to feel that team. The moment you pull the team away from the university, all those college students who play for the scholarship and love of the game are gone, because they're not going to give up their education and careers to be permanent minor league football players. And that means you no longer have enough players to field the team. So essentially, the university is providing to all those high-end players the competitive infrastructure to feel a team.
That's not even mentioning the stadiums, training facilities, support and medical staff, coaches, and all the other employees needed to keep that program running year-round, (important for this argument in particular) to help train, prepare, and showcase the high-end athletes for their chosen profession.
The real key is what the free market is telling us. If all those athletes were truly underpaid, then you should see a for-profit competitive league open up that would take all those players away from college football by offering a "fair" wage. It doesn't happen because the money would not be there to pay those athletes the money they want, and everything else necessary to field those teams.