Personally, I can't believe that any Patriot fans really believe this. I don't think that the OP is one, but it's a sentiment that I still hear around here.
Basically, I think that it's just an easy, simple, excuse to go to. It doesn't require a lot of though or analysis. "Patriots lost in the AFC Championship game - well if they had Branch they might've won!"
If you think Branch should've gotten his coin here, then that's fine - I think that is a valid and debatable topic. If you think that the reason the Patriots didn't win the Super Bowl was because they didn't have Deion Branch, you're delusional.
Generally, I believe that if you make it to the AFC Championship team, you have a championship-caliber roster. A lot of things can happen in the span of one game, that's what the playoffs are all about. The Branchless Patriots went up to a 21-3 lead in the first half. The Branchless-Patriots had numerous opportunities to put the game away late in the second half - they didn't do it. The offense couldn't convert a couple of big 3rd downs, and the defense played like it was a junior high team. Manning was absolutely transcendent in the second half, and the Patriots couldn't stop him or keep the ball long enough to neutralize him.
All that being said - they still almost won. So if you mean to tell me that if the Patriots convert that 3rd down with 3 minutes left and win, then it was a GOOD idea to not pay Branch $54 million, but if they DON'T convert that 3rd down then it was a BAD idea not to pay $54, then I think you don't really know what you're talking about.
Also, I love this aspect of Belichick's "ego". I'm not going to sit here and and say he doesn't have an ego, but Branch was let go (for a 1st round pick I'll add) because the Patriot front office (including but not limited to Belichick) believed that they could succeed without paying Branch all that money...well guess what, they made it to a very winnable AFC Championsip game. They were RIGHT. Just because they blew the biggest lead in conference championship history does not mean you should just reverse every personel decision they made.
Belichick has plenty of blame for this game. They were outcoached. If you want to ##### about personel decisions, you'd have a much more valid argument if you #####ed about the depth on the defense. Some of the names of the defensive players on the field at the same time in the second half of that game I had never SEEN before. Eric Alexander got OWNED all night. The lack of depth at LB and the secondary really hurt them, IMO - certainly a lot more than an upgrade at WR.

@ this team now holds the record for scoring the most points in conference championship history and STILL losing - and you come here saying they should've kept a WR. It'd be sad if this weren't such an obvious
BTW, I voted "No" on the poll.