I would submit that by brining in not one, but two deep threat WRs, Belichick would respectfully disagree with you that the lack of a deep threat WR wasn't a problem.
No doubt Belichick was hoping to have Branch play in 2006, but Branch was never really a deep threat (he sure isn't today)
As far as Belichick "moving" to a multiple TE system, Belichick has ALWAYS sought a multiple TE system - it wasn't until we got Gronk and Hernandez that he truly had the personnel to do what he wanted, but Belichick also knows that without a deep threat, defenses will collapse on his TEs - not to mention his short pass options and RBs, as well as put more pressure on his QB
Deep passes will never be the majority of throws - they are too high risk. But you need enough of them - a credible threat - to keep defenses honest to be able to make the small critical "must make"' plays.
I think Belichick knows those plays don't make a big difference in Fantasy Football stats - just in the final W-L stats.
As far as Stallworth, some try to make it sound like he was a failure here. Although he was not well utilized towards the end of the season (which I still view as a strategic mistake in an offense that focused almost solely on Moss and Welker - making those "must get" first downs more difficult in an offensive scheme that fooled nobody) it was Stallworth's $6 million contract in year two, and the clear emergency of Randy Moss as the better deep threat that made his cutting all but inevitable.
In 2007 Belichick may have disagreed with me a bit more than today, but not in 2012. Different year, different rules. When safeties could hit over the middle, the deep threat was more needed to bring pressure off the center of the field. Now it isn't. I still think he grabbed Moss because he was an elite WR for a 4th rounder.
As for his move to more of a TE focused passing offense, it absolutely has to do with the rules. Yes, he has alway preferred the two TE offense, but the rules changes have played into that favoritism. His two TE offense never made the TE the focal point of the offense like they do with Gronk and Hernandez.
All you have to do is look around the league and a lot of the biggest receiving stars are TEs (Gronk, Hernandez, Graham, Davis) or slot receivers (Welker, Cruz, Harvin, etc.). That is because the lack of contact is allowing the center of the field to be more open and teams are moving to more spread and horizontal passing than vertical passing. Again, making the deep threat less and less important.
Stallworth was a disappointment here. As the year went on he was demoted from #3 WR to #4 WR. I have read that he may have had problems grasping the playbook. Yes, his bonus was a part of the reason they didn't keep him, but if they truly wanted him back they would have tried to give him a long term deal to replace the contract. They didn't.
Besides, the Patriots never even tried to replace Stallworth with another deep threat. Tate and Price were never deep threat WRs although he would occasionally use them. Ochocinco even before they lost faith in him was rarely if ever used on deep routes. Lloyd probably ran more deep routes last night than the entire season combined. The Pats' deep threat receiver by design is Gronk.
I will conceed that in 2007 the need for a deep threat was far more necessary that today. I still it was overrated, but with the new rules for contact of receivers it makes the deep threat very much overrated unless you get an elite deep threat. Stretching the field may have been more important when a guy like Welker could get clocked every time he went over the middle. But defenders can't lay out receivers anymore changing the coverages. Less roaming the field head hunters. That is why teams are spreading defenses out and throwing more horizontally than vertically. Making an outside threat more important and a deep less so.
