The Leauge Office is; the teams aren't. The teams make all the money, so it doesn't matter too much.
But it still doesn't make much sense for even the League Office to be tax-exempt. It doesn't meet any of the general criteria of operation for the public good. It would be hard for the government to address, since that status has a lot of history and the league has a lot of money and goodwill behind it. And there's some complication with player pensions or something. The best scenario would be for the NFL to give it up voluntarily; it wouldn't be a high price to pay to remove a common criticism.
The "public good" criterion is for charitable organizations. The NFL League Office is a trade association that is tax exempt under a different set of criteria. (Of course you may disagree with trade association exemption criteria but they're pretty common--including within the software industry--as well as things like chambers of commerce. Or you may object to the specific exemption for sports leagues that's in the law.)
Update: Keep down-voting all you want. The Forbes article is weak in classifying all of the NFL as one entity. The below link gives a good overview of the structure of the NFL and it's tax exceptions. I'm glad I read it. That said the NFL still shouldn't be a non-profit organization.
http://www.todayifoundout.com/index.php/2014/01/nfl-tax-exem...
The NFL is an industry association; it exists to further the health and welfare of its industry (American football). Furthermore, the NFL doesn't take in revenue at all; it runs at an operating loss. Instead, NFL Ventures (which is a for-profit corporation) collects all monies related to the league and passes that money along--less a (taxed) retained cushion--to the teams and the teams' ownership, which are likewise taxed as per normal for-profit entities.
So, yes, your claim is misleading and your defense of it worse.
Almost. They exist to promote their part of the industry. They don't exist to support football generally and take many steps to put down possibly competitive leagues, such as using tricks to forbid players from moving to other leagues, even after their NFL careers are over. (Most commonly via delayed salary benefits that end should the player start with a non-approved league or if they speak out against the NFL.)
But it still doesn't make much sense for even the League Office to be tax-exempt. It doesn't meet any of the general criteria of operation for the public good. It would be hard for the government to address, since that status has a lot of history and the league has a lot of money and goodwill behind it. And there's some complication with player pensions or something. The best scenario would be for the NFL to give it up voluntarily; it wouldn't be a high price to pay to remove a common criticism.