Over the last 21 seasons, there are two things I've learned about Bill Belichick:
1) He puts the team on the field at the start of a game that he feels gives the Patriots the best chance to win the game, the only exception being when he wants to protect a starter or two for the playoffs...but even then he'd start Brady for a couple of drives or the first half in Week 17 even with #1 seed wrapped up.
2) He rarely makes mistakes in game planning and in-game adjusting...you can question a call or two he's made during a game (FG opportunity early in XLII, Fourth and Two...and a few others), but his record speaks for itself otherwise. Hard to argue with the best HC W/L percentage in the history of the NFL.
Newton, to say the least, has not set the League on fire this year. But he's also had the worst receiver corps (WR and TE combined), a marginal OL and an, at best, inconsistent backfield. He's never griped. He's been a Team player all the way.
I'm pretty sure that BB and Josh know what they have in Stidham.
In other words, while it's highly probable that the Patriots will lose this game, if they thought starting Stidham on Monday gave the Pats a better chance of winning, that's what they would be doing.