You said:
“All the sports leagurs also have some antitrust exemption.”
I don’t think that statement is accurate.
“In the US, three of the four major professional sports leagues do not have any general exemption from Federal antitrust laws. In a court challenge, all major leagues except baseball, would be subject to the antitrust laws under the Rule of Reason analysis, but generally not under the stricter per se rule. Because of this, leagues and clubs are able to counter an accusation of anticompetitive behavior with justification based on the special circumstances inherent in maintaining a professional league.”
https://digitalcommons.law.villanova.edu/cgi/viewcontent.cgi?article=1321&context=mslj
“To protect the rights of workers, unions were formed to negotiate employment contracts collectively in order to achieve a collective bargaining agreement (CBA). In the sports industry, the unions that represent the players are called players associations. Labor and antitrust issues are governed primarily by federal statutes.
Baseball, football, basketball, and hockey have all had legal battles involving the application of the antitrust laws. Baseball has held a unique exemption from antitrust laws in accordance with the interpretation of the Supreme Court in Federal Baseball Club of Baltimore, Inc. v. National League of Professional Baseball Clubs(1922). The Court held that antitrust laws do not apply to professional baseball.
The case, Flood v. Kuhn, (407 U.S. 258,) eventually went to the Supreme Court. Flood’s attorney, former Supreme Court Justice Arthur Goldberg, asserted that the reserve clause depressed wages and limited players to one team for life. Ultimately, the Supreme Court, acting on stare decisis “to stand by things decided”, ruled 5-3 in favor of Major League Baseball, upholding the 1922 ruling in the case of Federal Baseball Club v. National League.
Ironically, even though Curt Flood lost the lawsuit, the reserve clause was struck down in 1975. Arbitrator Peter Seitz ruled that since pitchers Andy Messersmith and Dave McNally played for one season without a contract, they could become free agents. This decision essentially dismantled the reserve clause and opened up the door to free agency.”
https://sportslaw.uslegal.com/antitrust-and-labor-law-issues-in-sports/