Just perhaps the Obama Administration has reminded them of that from time-to-time, which my explains the NFL’s recent PC turns.
Could be, but I don't really think they have to be reminded that much. Check out the political contributions of the owners and the commissar, and I'm sure you'll find they go predominantly to DEMPCRATS, with perhaps a few RINOS sprinkled in. That pattern goes back decades. The NFL is a very prominent example of crony capitalism, as are the other major professional team sports operations.
Heck, the NFL draft in and of itself - with potential employees coming out of college limited to employment with only one employer and being denied competitive market bidding for their services - seems like a blatant anti-trust violation, but, of course, the crony capitalists have gotten away with it for years.
Opensecrets.org is a good place to start looking. At least they record organizations/lobbying groups donations. And I doubt the NFL and teams for it wouldn’t be on there.
The NFL is socialism.