B/c with the current rule: A hail mary incompletion takes 5-8 seconds off. Spiking it gains 5 yards(you accept the penalty) and loses, 1-2 seconds max. If he were to throw a pick and the Giants run all around the field. He would have lost all of the time that elapsed and had one untimed play to win the game. It would have been smarter to spike the ball and save the time. I doubt Brady realized that, or he wouldn't have thrown deep.
For the record: If you do this (12 men on the field) two or more consecutive plays(although it doesn't HAVE to be consecutive plays), the Referee can rule Inpalpably unfair act and reward the offense with a touchdown. Thus you can't do it like 4 plays in a row and exchange 20 yards of penalties for 30+ seconds.
But I suppose if the team has no timeouts and are on their own side of the field, it wouldn't hurt to have 12-13 players on the field and hope the QB doesn't recognize this. (if he does, he simply spikes the ball. Loses a second and gains 5 yards). If he doesn't you hope your extra players get to the QB or break up the pass and make them waste tons of time.