Cross-posting but I hope relevant to this thread. Here's the houserule I'm considering:
* At 0HP the character is unconscious and finished for the fight.
* At the end of the fight make a single death save; success = stabilization, failure means the character bled out while everyone else was fighting.
* During the fight another PC can try to come to their rescue, e.g., dragging the character to safety to "stop the bleeding" with a healing kit.*
* Heal spells will not wake an unconscious/dying 0HP character, but will ensure stabilization.
* If the character stabilizes then roll on the Lingering Injury table (DMG p272). This injury affects the character for the rest of the adventure but disappears for the next adventure. Sleeping in the dungeon won't fix it, but downtime between games sessions will.
(Use of healing kit, bandaging wounds, CPR, etc, requires 1 minute, likely precluding use in a fight but not necessarily.)
Combining the above with healer kit dependency and slow natural healing.