What you mean is that the NCAA is not a state actor, so there is no 14th Amendment claim. That is how courts have ultimately ruled (often in the higher courts after opposite rulings in the lower courts), true. But it is only part of the story.
The courts have also ruled that the NCAA is a place of public accomodation. They ruled this just a few years ago (in Cuerton?), but the NCAA won the case on other grounds. This directly led to the NCAA changing their academic eligibility standards, however. In other words, state funds paid for the colliseum, and the NCAA gets to decide who gets to use it/play on it. That subjects them to a higher standard than a private entity. This is where the privilege/rights argument works against them. Use of publicly funded colliseums is a privilege, not a right, and if the NCAA doesn't like having to not discriminate (if they want to decide on whims, and not on legitimate interests) to use those colliseums, then it is free to play its games elsewhere.
Courts have upheld NCAA rulings and denied lawsuits based on the NCAA having a legitimate interest in making their rulings. These were based on student-atheletes who did not maintain academic standards. Here, what is the NCAA's legitimate interest? In those cases, they had their proof: the grades. Here, they have none. They DO have to prove their legitimate interest, that's the main point here. And without proof against wrongdoing on Sidney's part, then they cannot show they have a legitimate interest. They've never won a case without that.
In addition, all of the NCAA's court wins came back before the unification of the NCAA with the NIT. Those court decisions were based on colleges having legitimate alternatives to participating in the NCAA. With the NCAA now controlling all post-season play, that argument might not hold up anymore, especially since it was on shaky ground already.
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