Actually, it’s the anti-trust exemptions / lack of enforcement that allows the leagues (owned by the billionaire team owners) to decide who can and can’t own a team.
Otherwise teams could and probably would be purchased by publicly-owned corporations.
That's not a hard rule; there are several counterexamples. I found the Knicks, Braves, Blue Jays, Rangers, and Canadiens are publicly-owned. The billionaires will play their money games, but they don't have full control on the entire league.