I voted yes but have a small tweak.
You gain a level of exhaustion when you fail a death save (either rolled or forced).
When you drop to 0 HP, you immediately make a death saving throw of 10 or 1/2 the damage taken (whichever is higher). You can add your con modifier to this death saving throw.
I like this.
I would maybe not add damage to equation here.
also, if crit dropped you, gain one failed save and roll another.
when you suffer crit hit and do not drop to 0, roll a death save to avoid exhaustion level.
when your roll a 1 on any saving throw, gain an exhaustion level.
Exhaustion level:
-1 to every d20 check
-1 to all your DCs
-1 to AC
-5ft move speed per two levels(round down)
you die at 11 death levels.
you remove one exhaustion level on short rest and prof bonus amount of exhaustion on long rest.
optional feat:
Endurance:
+1 ASI
Advantage on death saves
Halve you effective exhaustion levels(round down, you die when you get 22 of them)
remove prof bonus of exhaustion levels on short rest and all on long rest.
Optional 2:
take exhaustion for benefits.
Suffer on level of exhaustion to get Heroic inspiration
Action surge, Action: take additional Action this turn.
Suffer 2(or maybe 3) levels of exhaustion.
You can do this only once per turn.
You need to take a short rest to take action surge again.
Action surge, Bonus:
take additional Bonus action.
suffer 1 exhaustion level
once per turn only.
recover a spell slot. Bonus action
1 exhaustion level per slot. max 5th level spell slot.
second wind:
Bonus action,
suffer exhaustion level
use a number of HDs to heal yourself up to your prof bonus.