The anti-trust exemptions that have been provided to all the professional sports have to do with protecting the franchises from charges of restraint of free trade based on the manner in which they limit the mobility of labor. There have also been questions of anti-trust litigation over time from cities that wanted a franchise but were not granted one.
Professional football actually existed for over 20 years before it was organized into franchises with the creation of what subsequently became the NFL. However, the sport did not thrive because there was no way to structure effective competition among the teams.
In a similar manner, our economy does not realize the advantages of a market economy in the absence of competition. Accordingly, the greatest involvement of government in business is to prevent business practices that stifle competition. That is not socialism; it is the protection of capitalism.
I have no problem with the way that professional sports are organized in our country. I simply found it ironic that this thread was using the NFL to illustrate how socialism might work if applied to the sport. The irony is that the whole sport is based on a socialistic redistribution of resources from the stronger teams to the weaker teams.