The "Sit Monahan until the deadline" argument is flawed.
So they sit Monahan (~3rd best player, annoying the players on the team).
31 other GMs see that, and think "Gee, I wonder why they are sitting this guy two months from the trade deadline? They must be pretty concerned about the likelihood of injury to cripple their team that much. Let's wait a bit and see who's available in March."
Then what? You just wait, for weeks and months waiting to see if a GM calls you on Monahan? What happens to his value as he sits out?
Yeah, Monahan is an injury risk, and yeah in a perfect world they'd move him early to reduce the risk, but they need a good offer to make a trade. It takes two to tango.