Earlier quoted context omitted.
The US Football league NFL also has an antitrust exemption so is deserving of public scrutiny.
I don't believe this is correct, which is why competing leagues pop up every few years. You might be thinking of Major League Baseball?
Baseball has an exemption due to a court case, though their hiring cartel was broken in the courts, resulting in free agency (same for NFL, with the Roselle(?) case).
But the important one is all about the money: monopoly control over monetization of broadcast. I direct you to 15 U.S. Code § 1291 which you can read for yourself at https://www.law.cornell.edu/uscode/text/15/1291