I agree to a certain extent that winning the SB alone can be enough to quell people in general, but it's also worth noting that Coughlin, by the admissions of several NYG players, had actually worked to change how he interacted with his players which in turn led them to respect him more. I always thought Cowher was a little bit overrated as a HC and still do. The SB did not change my mind. I believe that the quality of his OC and DC had more to do coaching wise with why they were in the SB than he did.
Expanding on that idea, Coughlin also knew when coordinators weren't working out. He fired Tim Lewis. Yes, his team made the playoffs that year, yes, they even made it the year before, but the fact that Lewis's defense wasn't working out ultimately held the team back and Coughlin knew an improvement needed to be made.
This is all moot because Trgovac left on his own terms, but Fox was ready to keep a coordinator whose defense was arguably one of the worst the last 8 games of the season, probably just because they made the playoffs.