Oh gods, not this raising it's head again. There's multiple threads why (a) exhaustion in specific is the worst choice for this because getting a level of it gives non-combat-primany penalties and makes you want to make everything combat -- the wrong penalty for going down in combat, and (b) the characters most likely to go down are front liners, often who make character choices to protect others. So penalties longer then the combat scene are penalizing someone for team play and sacrifice that we instead should be encouraging.
Basically, it's the wrong mechanical choice and the wrong design choice. Like, the exact opposite direction we want on both of those aspects. Yes, Exhaustion is an underused tool that's easy to reach for. But giving it any consideration in terms of mechanics or play shows it's not just the wrong tool for the job, it gives the opposite effects of what we want. Give some other penalty, that makes people hurt not want to get into combat, and doesn't penalize a standard and selfless mode of play (tanking) past the scene that it occurred in.