Great post.
However I do think the draft can help the pass rush/pressure situation. Sergio Kindle should be available at 22 and Carlos Dunlap should be available in round 2.
Some really good WR's may also be there in rnd 2, such as Brandon Lafell, Damian Williams ,M. Gilyard, Demaryius Thomas, Arrelious Benn, Desmon Briscoe,
Jordan Shipley and Danario Alexander. One or 2 of these could give Brady weapons the way the Saints have weapons. Imo, the fastest way to improve this team is by giving Brady the weapons so he can spread out his passes. With only Welker/Edelman and Moss as legitimate receivers the Pats were too easy to defend in critical situations.
I'm not ignoring other needs such as LB, DE and TE, but, as you said, the D wasn't all that bad. And with BB putting his reputation on the defense, I think it will improve, as you do.