Corporations are taxed on profits. A non-profit entity isn't taxed on profits because it's expected that all profits will be re-invested in the organization. The NFL league office does run a profit every year because inlays exceeds outlays. The profit is only about $100m. They distribute some of it back to teams (which is then taxed) and then use some to pay overhead and salaries.
My paycheck isn't $44m. Compared to the corporate tax rate (which would be paid by the league office itself, and not the employees of the league office), the personal income tax rate - even for the highest earners - is still lower. This is largely because of the regressive nature of payroll taxes, which stop after $117,000 (which Goodell hits towards the end of the first day of the fiscal year). If not for the tax-exempt status of the league office, Roger Goodell would simply not be making $44 million; the same could be said for many non-profits, where the executive and board members would not be making such a high salary if not for the tax-exemption.
Ultimately, I'm playing devil's advocate here because I don't expect the NFL to do anything more than it has to in order to protect its image, and the cheaper the better. Such is the nature of capital.