The more I think about that the less I like it.
It doesn't actually solve the problem as you still want to cure the character asap anyway.
And it's just not fun. Not only did the character go down and you may miss a turn or two, but now the character is bad. This lasts at least a long rest, maybe more.
Which is why in my own homebrew suggestion, I made it temp. exhaustion which goes away when the fight ends. Essentially, the magic that restarted your heart hasn't had time to regulate the beat yet, so you're pushing too fast too early.
After the fight and the adrenaline has gone done, you stabilize and lose the exhaustion. That way you are only penalizing players during the current fight, and they can start dropping dead instantly the sixth or seventh time you heal them up.
To answer @
Chaosmancer specifically and succtinctly:
The reason heal-from-zero is an issue now but wasn't then, is that 5th edition is the first edition where all damage past zero is simply not counted.
This opens up a previously unavailable healing strategy where you (ab)use the way negative hp doesn't count to provide the bulk of the healing for you.
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This means that if a huge monster deals you 40 points of damage, if you had only 5 hp, 35 of those damage points simply evaporate. This is a HUGE savings in healing oomph.
Compare to if you had 75 hp. Then you'd take the full 40 damage and fall down to 35 hp.
Traditionally, a cleric assuming the role of combat-medic would cast a heal or cure spell from a high level slot to heal all those 40 damage.
But in 5E, the cleric can choose a different strategy. If he waits until you fall to zero hp, he can then cast a cheap Healing Word spell (cheap in that it only spends a first level slot; cheap in that it only requires a bonus action; cheap in that it can be cast from a distance so the cleric doesn't waste time moving back and forth on the battlefield).
This will bring you back up from unconscious, with a few hit points. Effectively 35 of those 40 incoming damage points are simply gone. And the best part is - the next time the monster hits you, 35 of those 40 damage points are negated once again.
Each time you the Cleric
1) save movement
2) save spell slots
3) save time (since you only spend your bonus action, you can use your regular action for offense, to help defeat the monsters)
To talk around to my point, the major issue does not seem to be the heal-from-zero then.
Hear me out.
This stems from the desire to have the most efficient use of action possible, no matter what. In games I run (which tend to have less experienced players) getting hit for that initial 40 damage causes panic. The healer is being begged to heal the character, the damage dealers start seeing if they have any tricks left, the thought of retreat might even possibly enter their mind.
The idea of not healing, so that later healing could be done at less value for less wasted resource does not enter into their equation.
Now then, looking back at the OP's proposed solution. What is the most efficient use of the Healer's action if they want to still be combat capable and do the least amount of healing necessary?
They don't heal.
Even if you make that 1-3 minutes count as concentration so they can still do other things, the type of people who are analyzing the most efficient use of their abilities and time are not going to drop a Spirit Guardians or Bless to go over and try and heal Bob, even for 60 hp, at the cost of that being the rest of their fight. And they will do the analysis that the possibility of them getting hit (since half the time they will be next to the monster that killed Bob) and losing that concentration and therefore their spell, is very high.
So, if the issue is that players who are looking for the most efficient way to heal are willing to key party members "get killed" to not have to heal 35 points of damage, the original solution just means they aren't going to heal period, because it is a waste of their time and actions.
Tracking the negative hp could potentially be a better solution, because then they need to use more powerful magic, however, if a player is down to -35 hp, you'd need to heal them for almost 70 pts to get them back in combat shape, and healing spells don't get that powerful for a long time in game, so then you are expending approximately double the resources to get them back in the fight, and with rolling for random healing, you could still end up healing them for 0 effect, wasting a turn, and they may decide to let a character die, and then use a different ability to bring them back (depending on level and wealth) instead of wasting multiple slots trying to get enough curative magic fowing into them to get them back into fighting shape.
At least, that's how it is looking to me