The way I look at it is 1st you go to your bread and butter and that's Welker, first mistake. The bigger mistake was trying to stop Indy from scoring! Why not let them score if your gonna go for it on 4th down, this should have been apart of the decision. You let them score and now you have almost 2 minutes to get in fg position to win the game. You have Brady, Moss, Welker and Watson to get you there and then you have one of the best FG kickers in the game.
To me that was the biggest mistake!
It's a tough call on letting them score. I understand the point, but they make one mistake and you win. I would not have let them score initially, or in the final seconds once it was obvious you wouldn't have enough time. The one play they might have allowed a score, and have it be the proper decision, was the running play that busted down to the one. There was still more than a minute. But New England has stopped the Colts before on a goal line stand to end the game. I'm sure Belichick remembered that. That was in Indy, somewhere between 2003 and 2005, another high scoring game. My memory seems to be less reliable because normally I could pinpoint the exact season.
One strange variable is the correct thing to do can change during the play itself, depending on the situation. I don't know how you school players to grasp the shifting circumstance. For example, Miami led Oklahoma 21-20 this season, with Oklahoma desperate to get the ball back late, with no time outs. Miami made a key first down on a 3rd down play. Once the first down was made, the correct strategy would be let the receiver score and make it 28-20 so you still have a chance. But how do you tell a guy, tackle him on the 27 but once he reaches the 24, let him score? Tonight was similar. The Patriots secondary probably should have allowed the back to score once he burst down to the 1 yard line, but if the play was defended properly near the line of scrimmage, then you're better off stopping him.