Carroll provided some context around that last play, although I don't know if it really washes. They had extra receivers against NE's goal-line group, they had 3 downs, 1 TO. So they were going to do some combo of 2 runs and 1 pass with the remaining 3 downs.
It makes sense that you have some combo of 2 runs and a pass for the 3 remaining downs, but THAT throw does seem problematic. It's going to get HotTaked to death because the worst possible outcome occurred, but the chances of a pick there ARE pretty nil.
It may have made sense to run inside, TO if it doesn't get in, and then run a Wilson bootleg on 3rd down with a releasing TE. If the TE gets open, you have a pretty easy, low-risk throw. If it's not there, you have one of the fastest QBs in the league who can probably jog to the pylon. If someone does get a bead on him, maybe he gets knocked out and the clock stops. Even if he doesn't, you still have time to reset and try to plunge it in on 4th and short.
It's a bad call throwing into a tight spot, but I think the OMGZ YOU JUST HAND IT TO LYNCH!!11!!! analysis is a bit too simplistic.