You are quite mistaken. You have a source for that?
The NFL has revenues exceeding $4B which are nearly all "profit" (that is, revenues>opex). This cash flows through the league and then is distributed to the teams, who all of which run on essentially zero taxable profits. In other words, the revenues of the teams are the "profits" of the NFL non-profit. That is why there is no need for profits--because they distribute the cash out as 'expenditures'. The level of expednitures is aribitrary[1]...it is just set to maximize the flow of funds from the tax-exempt body with the least amount of tax.
At the lower level, the Teams are structured to run on nearly zero profits. This is why they can-not "afford" to build stadiums of pay taxes etc. This of course is also a highly-structure tax-avoidance setup, where inflated costs ("sweetheart" deals) are made with affiliate entities, which shelters the teams from paying taxes. The locations and corporate business structures of the "related entities" are likewise set up to be both opaque and tax-optimized in legal structure and jurisdicion.
The values of NFL franchises are staggering. Their debt-loads are also staggering. Hint: the banks would never lend them this money if they did not know more than your average punter. In part these investments are "ego-polishing" (like the America's cup), but there is quite a bit more money floating around the NFL, and this money is going to make rich people even richer...very little of it is being paid in tax.
[1] There is no "true sale" of the NFL non-profit's product; they take no risk in holding inventory. Their main purpose is to shield the teams from 2 things: (1) anti-trust; and (2) tax. To wit: a tax-exempt monopoly that makes no profits? Something should immediately trigger your sense of the non-obvious being present.