Bill won 2 Super Bowls as the Giants defensive coordinator back in the 80's. There is no "need" for him to win another to cement any legacy. That legacy is cemented in place and he will be the greatest football coach ever, when it's all said and done.
That's not to say, with his competitive fire he doesn't want to win more titles, but I am not sure it's driven by some need to prove he was more instrumental in the Pats' success than Tom Brady.
As for who was more instrumental to the Pats' success, it's a chicken and egg conundrum. I think it was both of them, regardless of what Tom does in Tampa.