Oh, hi, hello. I am more than willing to share what we came up with. (The “we” here being @gloomdraws and myself. And by “myself” I mean “it was mostly Gloom and I just sort of sparked the conversation.” You should ask him about D&D. He’s super good at it.)
Modified Dying Rules
If damage reduces you to 0 hit points and fails to kill you, you fall prone and are Dying. You remain prone for as long as you are dying.
At the start of each of your turns while you are Dying, make a death saving throw. Roll a d20, if the roll is 10 or higher, you succeed. Otherwise, you fail. A roll of 1 counts as two failures. On a roll of 20, you regain 1 hit point.
On your third success, you stop Dying and become Stable.
On your second failure, you fall Unconscious.
On your third failure, you Die.
On your turn, if you take an action or bonus action, you must make additional death saving throws after the action resolves, with disadvantage.
If you become Stable or restore hit points, you are no longer Dying and your successes and failures reset.
Stable-
On your turn, if you take an action, a bonus action, or move more than half your speed, you must succeed on a DC 10 constitution save or resume Dying. The DC increases by 5 with each success, and resets if you regain hit points.
The summary is that you do not become unconscious until after you’ve failed two death saves. This means you can get dropped to zero, then could potentially take one more not-crit hit that round and still have a chance to get yourself back in the fight (or extricated from it if that’s your choice). Or you can opt to spend your whole turn attacking, but at extreme risk to yourself.
It’s helpful for the player who hit zero so they don’t have to rely on someone else getting them back up, and also relieves some of the pressure of definitely requiring a healer to get people back in the fight. (Or that’s how we feel about it.)