Don't confuse "nonprofit" with "charity". While virtually all charities are nonprofits, not all nonprofits are charities.
The NFL being nonprofit is simply a reflection of how the league is organized and equity and earnings are allocated. In this case, most of the equity in the NFL is held by individual teams and the teams' billionaire owners, and all the earnings are targeted to those same teams. The league acts as just a vehicle for the teams to coordinate functions like marketing, scheduling, and league matters. So when the league gets $10 billion in TV contracts, all the profit is distributed to the teams, which then pay taxes on it. Being structured as a nonprofit, the NFL league has trustees and beneficiaries. It could reincorporate as a for-profit, in which case it would have owners and shareholders. In that case, each team owner could be granted one share. If that were to happen, Paul Allen instead of receiving one tax bill for $100 million for the Seahawks, would get two tax bills for $70 million (for the Seahawks) and $30 million (for the NFL share). From the taxman's point of view, it's pretty much the same.
There's nothing sneaky about the NFL being a nonprofit. It's just reflects how the league was originally set up.