Death saves, as far as I have been able to conceptualize them, represent a range of possible injury levels, and the uncertainty experienced by the players and characters as to which has happened.
If I roll a 20+ on your Death Save, clearly it's not possible that my character's carotid was bleeding out but suddenly re-knit itself. I think regaining consciousness spontaneously represents having been dazed and knocked out momentarily, but it not being that serious an injury.
If I roll 3 successful death saves my character stabilizes- again this must mean that the injury wasn't actually critical/mortal, but was enough for shock and pain to knock him out. But he's not bleeding out. He'll wake up on his own like an action movie hero knocked over the head and left for dead, and be able to get back on his feet after 1d4 hours.
If I roll 3 fails my character turns out to have been badly injured, and bled out without anyone coming to save them.
There are a large variety of possible injuries which could constitute the second or third situations, and I don't know if it'd be feasible to specify all of them with contingency spells.