You need to go back and watch those Super Bowls, because in 2004 the Patriots ran the ball for 80 yards, threw it for 350, and Brady set the Super Bowl record for completions, as he went back to pass 50 times, and the defense also gave up 325 passing yards.
That 2004 Super Bowl champion is NOT any different than the current scheme. We may not have the players, but we were a diversified passing offense back then, with Antowain Smith at RB, and he was running the ball at an amazing 3.3 yards per carry.
The key is that all 3 Patriot's opponents respected the run just enough to open up the passing game. Just enough...