To quest
The whole sequence was a mistake. They had the ball with 2:11 left and passed the ball incomplete on 3rd and 2. They then called their last TO before they went for it on 4th and 2. That left them with no timeout and still 2 minutes on the clock and no way to challenge the spot on the completion to Faulk. Belichick doesn't make many mistakes but he did make one there.
To question Bill's decision is fine, but to, with crystal 20-20 hindsight as a "mistake" is simply lazy thinking. There were a whole host of factors that went into that decision that I can think of just from my own remembrance of it. let alone the other possible factors that as mere fans we weren't privy to.
Contrary to revisionist history, the Colts offense WAS having its way with the Pats defense as the game wound down. Bill would have had a much better idea of just how his defense was looking to him than DI in his august wisdom sitting in front of his TV. Maybe it not what DI thinks, but we all know he thinks he's smarter than Bill for starters. But I recall that the Pats had contended with an outbreak of the flu that week. That the dome "for some reason" seemed to be inordinately hot as the game went on( along with the piped in crowd noise).
So here is what Bill was faced with. He could come close to ending the game by gaining just 2 yds. or he could punt it back to Colts and likely give them good field position going against a tiring defense weakened by the flu against one of the best offenses in the league at home. So again, does it seem so "outrageous" for Bill to think he had a better shot at getting 2yds than holding off the Colts. Our offense didn't suck BTW.
The same goes for Shanahan's thinking. First of all, 46yd FG's are NOT gimmees. A lot of things could have gone wrong. 2nd Shanahan was pretty unlucky. His players didn't exactly help him out. He knew that HIS defense was on its last legs. He knew his run game wasn't going anywhere. So Ryan took a bad sack. But if his LT had executed and not so obviously held he would have gotten back to at least long FG range. Was it such a bad call or just bad execution?
Same goes for Seattle game. That was not NEARLY as bad a call as it has been made out to be. The Pats were in a defense that gave them a clear advantage vs the run. Not to mention the fact that the pick play they designed for man coverage the Pats sent in was a good one. It took great execution by Brandon Browner as well as Butler to pull off that miracle. Sure the Pats might have stopped it, but to pick it off was nothing short miracle. No on would have predicted THAT. It was tantamount to a one outer in poker. Butler make a strong move to the ball, but the collision was such that it knocked the receiver on his ass.
So on all of these it is fair to second guess the calls, but NONE of these calls were "mistakes" but rather just one of hundreds of decisions coaches make during the course of a football game. All made with a LOT more info than we fans. the media or "experts" have on hand, and like all decisions, some work out and some don't.