I agree with you that Belichick is obviously a man who is going to play the highest percentage, and in many of these cases, he doesn't want to risk losing a timeout. In the first quarter, second quarter, even third quarter--that's fine.
However, when you have 2 or 3 timeouts remaining at the end of the game as the other team is marching down the field to get themselves into a potentially game-winning FG attempt, you take the timeout if all else fails.
He and his staff had to have known that it was a borderline call. Since the Jets player was not in bounds, their drive may very well have been stopped without even getting that FG attempt off--and that is the obvious point. There's nothing to lose. You aren't collecting timeouts to take home with you. We all know that Belichick wants to win and put the team into the best position to win the game, so it's got to be an oversight, or a bad system that they have in place.