I tend to agree with you, that constant pressure is due to the lack of a running game. Brady has always performed under a balanced Offensive game plan. The 3 SB years, they have averaged 50-50 pass-runn, in attempts. That is a well coached and well excuting team. Now what did they do in 2002? They went away from the run, 60-40 pass-run, in attempts. They finished that year with 9-7 record (56%).
Under AJ's start, there was very little attempts in the running game, basically passing to keep up in scoring. The Offensive game plan was 62-38 pass-run, in attempts. When you pass that many times, the defenses don't respect the run and the passer has to be perfect in read and execution. That was something that AJ could not be last year, with that porous OL (52 sacks for the year, I believe).
Anyways, my point is if you have to put the ball up 62% of the game (AJ), compare to 48% for Brady, you will have more chances of making mistakes.