It's hard to know Bill's actual thought processes here, so we can only speculate based on past words and actions.
Bill (and the coaching staff) probably figured that with a limited Cam Newton they were able to get to 6-6 with an outside shot at the playoffs after winning 4 of 5. Of the QB's on the roster, the coaches felt that Cam gave them the best chance to win any given Sunday. As hard as that may be to believe, it was probably an accurate based on what they see in practice. Belichick does not believe in the "game day player" nonsense, so Stidham obviously hasn't shown anything in practice. However, they do need someone to take snaps in case Cam goes down, so Stidham and Hoyer remain on the roster.
At this point, it's hard to see what the Pats have to play for. The slim mathematical window to reach the playoffs will likely be closed after the weekend. So there could be benefit for the team to see what they have with Stidham. However, the coaches may already know the answer, and it's probably something we may not want to hear.
Finally, Cam is respected veteran who by all accounts has earned the respect of the locker room. Belichick has never openly thrown a player under the bus after a tough loss, and that includes Adalius Thomas. There's no upside to him causing a QB controversy with vague statements in the post-game presser. Drew Bledsoe was mentioned upthread, but he was hurt when the change was made, and he communicated his eventual decision internally before going to the press. If he's going to make a change, he's going to talk to Cam first in private. Then he'll to the press that he's changed his mind and is making a move in the best interests of the football club, and live with the mediots laughably slinging "hubris and arrogance" at him.