I think I can understand the NFL decision from a different viewpoint. The NFL is a smoothly run big business, and the games are a big deal. There's a lot of security planning, in fact the NFL has its own security division (which I presume works with the local stadium venues to ensure that there is a comprehensive security & emergency plan in place for games - coordinated with the appropriate local LE agencies).
So if there is an incident (large or small) at a game that has to be handled in accordance with the security & emergency plan, do you really want an unbriefed, armed & possibly intoxicated off-duty LEO - whose intentions are unknown - wandering into the middle of the incident to do something? And by intentions unknown, I don't mean he/she's contemplating criminal action, I just mean planning to take some LEO action that's not unreasonable on it's own, but is unbriefed and throws a monkey wrench into the pre-established security and emergency plan. That has a real chance of not turning out well.
And yes, armed LEO's are welcomed on airliners - but that's really different. They must check in early with the customer service agent & have their credentials verified before they are allowed through the security checkpoint. They are boarded early and all armed LEO's on the flight are advised of each other's seating (as well as any prisoners they may be escorting) so they will know if there are any other armed LEO's on the flight and where they are seated. They are also prohibited from consuming alcohol on the flight. Also, an airline flight is a much smaller venue with, by definition, a much lower number of participants, usually a shorter duration, usually a much lower emotional intensity than an NFL game and arguably a lower likelihood of a major event.