I think you are (intentionally?) missing the most obvious reason for not risking it which is not making his condition even worse. He could have aggravated this issue and landed on IR instead of being a proven backup in case Brady is out for a game or two.
I don't know how that could be obvious.
I've never seen Belichick take the approach that I am not going to play someone who is able to play because I don't want them to get hurt. Further, I'm not sure why you would start your 3rd stringer in an actual game because starting your second stringer might mean that you have to start the 3rd stringer in the future in the event both the starter and backup are injured.
The argument would amount to this:
I have a division game today. We are 3-0 they are 1-2.
If we win today, we bury the entire division 2 games behind us.
If we lose we only have a 1 game lead, and lose the tiebreaker, even worse with a home loss.
Jimmy G has played great and can run the entire offense.
Brissett is raw, and we can trust him to run the entire offense.
If I play Jimmy G we have a better chance to win, but if he gets hurt and then Brady gets hurt there is a longshot chance that I may have to play Brissett in a future game that more likely than not, wouldn't be be against a division opponent who can get within one game of us.
So I'm going to play Brissett in order to reduce the chance that I have to play Brissett.