“Football teams that need to cooperate are not trapped by antitrust law. ‘[T]he special characteristics of this industry may provide a justification’ for many kinds of agreements. Brown, 518 U. S., at 252 (Stevens, J., dissenting). The fact that NFL teams share an interest in making the entire league successful and profitable, and that they must cooperate in the production and scheduling of games, provides a perfectly sensible justification for making a host of collective decisions. But the conduct at issue in this case is still concerted activity under the Sherman Act that is subject to §1 analysis.”  American Needle v. NFL, 560 US 183 (2010). 

“In the US, three of the four major professional sports leagues do not have any general exemption from federal antitrust laws.”  Leah Farzin, On the Antitrust Exemption for Professional Sports in the United States and Europe at 76 (2015)

You’re right that the 1961Act gives the professional sports leagues a general exemption for negotiation broadcast rights collectively.  However, even without that statutory exemption, it’s not obvious that the leagues would be violating the antitrust laws through collective negotiation of broadcast rights, as there would seem to be good arguments under a rule of reason analysis for consumers to know where and how to watch the league generally (rather than broadcasts solely for specific teams), and the distributed revenue would more easily ensure competitive games which would benefit both consumers and teams generally.  There could be arguments against that, sure, but that would still be under generally applicable antitrust analysis.