Joe, in your very specific scenario, the player should volunteer to take himself out. Because you said the player is "absolutely positively certain that the backup QB would give your team a better chance to win".
However, I think in real life, this point of your scenario is often the issue. I think often an injured player, especially one who is viewed as significantly better than his backup(s), will typically think that he offers the team the best chance to win if he plays through the injury, providing he is able to perform the basic functions of his position. I think this mainly happens because of the combination of self confidence and competitiveness, as well as possibly because of the perception of the backup's ability (or lack thereof).
Because of this, I think it is typically the responsibility of the coaching staff, with input from doctors/trainers as needed, to make this decision for the good of the team, without expecting that the injured player will necessarily volunteer to come out if he can still go.
It sounds like the coaching staff made the decision for Cutler yesterday, which is appropriate if it was severe enough. It's unclear exactly how severe his injury was... it's possible that despite the normal desire to play on, he knew after the first series of the second half that the injury was so bad that he couldn't perform well enough, so he did not fight them. Unfortunately for Cutler, there was no way for the viewing public to tell that his injury was that severe, since he didn't seem impaired on the sideline. Alternatively, perhaps it wasn't really that bad, and it was still the coaching staff's decision to take him out, presumably to protect him from further injury. In that case, we might have expected to see him trying to get back in, showing emotion, etc., but we did not see that. So either way, what was visible to the viewing public didn't seem so support him being taken out. That led to a lot of the criticism IMO.