How does the Department of Educational "enable" the NCAA to have restrictions on paying players? Just by not forbidding it? That sounds like classic hardcore leftism - equating a lack of government regulation to government support. And by the way, the NCAA system existed decades before the the Department of Education even existed.
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.