Forget Moss in the '06 AFCCG, if they just had Branch there's no way they'd lose.
I don't think he would've sucked in NE, nor been so amazing that he'd have saved Pete Carroll or Drew Bledsoe's job. But I think it's unlikely he would've stuck around under BB any longer than he did in real life. '07 was amazing, '08 was fun but...y'know, '09 was transitional, and by the time that transition started to manifest itself in '10 he didn't even make it to October. While the Patriots era through the Moss tenure was strange, it was hardly a poor situation for a superstar WR with a high football IQ. The offense was being re-tooled but he was always going to get his, he was playing with the GOAT QB and HC (and he knew as much), and could've helped that team in '10 in particular over the top. The better angels of his nature just only hold out for so long.
Moss ended up in the right place with Dennis Green and Cris Carter. I feel ridiculous saying it in retrospect because those guys (RIP Denny) kinda come off as douchebags but he toed the line and thrived as a football player while that structure was in place in Minnesota. Once it was gone and he had Mike "Randy Ratio" Tice in place and Carter was retired to Miami, Randy was supposed to be the veteran leader and the wheels eventually came off. That being said, the "mooning" in Green Bay that probably cemented his way out of MIN to OAK and then NE led to maybe 2 of the best NFL moments ever - 1) Joe Buck's lame outrage as the scene unfolded, and 2) If you don't write checks how do you pay these (NFL) guys, Randy?: "Straight cash, homey". And I will always love the guy because he was as fun to listen to and watch off the field as he was mercurial on it.