The NFL is a nonprofit by any reasonable definition, even without the explicit exception in 501(c)(6). Or rather, it was, until it voluntarily gave up its tax-exempt status in 2015 supposedly over public opinion about their status.

I think people simply aren't aware that all the major revenue associated with the NFL goes through specific teams, which are normal taxable corporations. Humorously, people also aren't aware that they gave up their tax exempt status, so I'm not sure that decision was worth it.