Frankie1969
Adventurer
The exhaustion is to make sure this isn't a free pass if you know you won't die (or stay dead long).I really like this, though I would drop the exhausted condition as the failed death save is a pretty high cost already and you've already halved their speed. I would also skip the disadvantage on death saves since presumably they would only do this if they want to take an action or reaction, so they are automatically failing the save anyway.
The disadvantage on death saves is to make sure this isn't a free pass compared to Unconscious.
If your wounds haven't stabilized yet, you really shouldn't be moving around.