That was an opportunity for Goodell to show...INTEGRITY... and end this, even if it meant firing some of his closest advisers/friends. He could have told the truth, that Kensil, Vincent and company acted over zealously and were wrong in their assumptions.
Had Goodell done that, he would have been praised for taking a stand and making the right decision, even if it made him look bad and led to the dismissal of some of his inner circle. I don't believe it would have led to his dismissal, because, if he survived the Ray Rice debacle, he could survive this. Even if it did, it wouldn't have been immediate and Goodell could have engineered a nice soft and lucrative exit, while having a say in who his successor is.
As far as Goodell having some dirt on New England, why wouldn't he use this information to get Brady and the team to back down, accept things, and drop this? If he's not going to play that card now, when would he? I think it's simpler: Goodell doesn't believe "his guys" were wrong and he definitely doesn't believe that he could ever be wrong.
As I said earlier, Goodell could have made a tough call and ended this all sometime between January and May of 2015. Instead, he chose to protect his cronies and himself. In the long run, that decision will do more to damage the integrity of the game than 11 footballs below 12.5 PSI.