This was an interesting example of Rex Ryan's approach to coaching.
This was a preseason game against the local Super Bowl winning team. Rex really wanted to win.
When Geno went down, rather than go to the next guy on the depth chart, Rex put his likely starting QB in late in the game. Never mind the injury risk behind this offensive line - not only second/third stringers, but also a lineup that Sanchez had not likely worked with at all. Never mind what getting put in during the fourth quarter could potentially do to a demonstrably sensitive guy.
This is a very different approach than Belichick would ever take. It speaks to the emotional, competitive nature of Ryan. He wants to win so much that the fourth quarter of a preseason game becomes so important - to win, not to coldly evaluate the back end of the roster. Belichick has a plan, and he is going to stick to that plan. Ryan is a little more James Tiberius Kirk, but without the scriptwriter saving him. You can see how the emotion and passion stirs the team, especially in the playoffs. There are times when that passion translates into performance on the field. And there are times when that passion just backfires. Just as there are times that Belichick's less emotional engagement make the team seem like a colder place.
Brady is obviously a different context, compared to Sanchez and Smith. But I can't imagine Belichick doing that to, say, Chandler Jones or Solder.