I feel like what the rule is saying is that "could kill him" means, "since you can't roll again, if no one else can and it's a lethal injury, you've condemned them to death" (i.e. you should push the roll)
In the spirit of the game (and the explicit section "The Art of Failure" on page 44), I make sure not every person decides to roll off the bat. If the Enforcer with 2 Empathy and 0 heal wants to roll, he can - BUT when he fails, the dying person is going to take a trauma in another stat. Or maybe it adds 1 (or even a d6) days to the critical recovery length. So preferably, the Chronicler with 8 dice should be the one doing the roll. If he fails, then the next best guy, etc.
What this does it keeps every single person from wanting to roll on every single skill check. This is good for the speed and simplicity of the game.
The most important thing a Mutant GM can remember is "Failure should always have consequences."