Kinematics
Hero
Another idea that may be a little easier to work with would be that dropping to 0 HP puts you at level 6 Exhaustion (ie: dead). This would be a short-term/temporary version of exhaustion, and doesn't overwrite or remove any real exhaustion levels you may have.On this, however, you have my absolute support and agreement!
And if they don't bring back negative h.p., another answer is some sort of rule or system that severely limits what you can do for the first few rounds after being cured up from 0. Maybe a scaled and unmodified die roll, something like:
1-2 - you remain prone and cannot rise. You are defenseless.
3-5 - you remain prone and cannot rise; you get the benefits of your defenses, and attacks against you have advantage. You can do nothing other than defend.
6-10 - you can move at 1/4 rate, all attacks are at disadvantage, melee attacks against you have advantage, you have no reaction or bonus action this round, and you cannot cast spells of any kind (but can activate or use devices)
11-15 - you can move at 1/2 rate, you can cast a spell or activate or use a device, but you have no reaction or bonus action this round
16-20 - you are fully functional and can act as usual.
So if you got cured from 0 you'd roll against this table each round, at +1 for each previous roll, until you got into the 16-20 range; and a subsequent roll cannot make your situation worse (e.g. if you roll 12 in the first round of recovery then roll 1 in the second round, the 1 is ignored as your situation cannot get worse) until and unless you get hit back down to 0 again, which starts this process over.
Note this is an off-the-cuff shot at this; were I designing it for real it'd be a lot more granular - probably a different condition for each number below 16.
Healing effects (spells, potions, Second Wind, etc) reduce the exhaustion level, usually by 1, but possibly more when using high-level effects. And once you're no longer at 0 HP, your temp exhaustion also drops by 1 point at the end of each of your turns.
This means immediately after a simple Healing Word, you're sitting at speed 0 and prone, unable to move or stand up, and at the usual disadvantage. Next round you can move at half speed, but your max HP is halved (potentially also limiting the value of using Heal on a 0 HP person). And so on, down the exhaustion table. You can start getting back into the fight within a couple rounds, but you'll likely be struggling. Having a healing potion on hand is the best way to get back on your feet, as you'd be able to drink that even at exhaustion level 5, and it would be another -1 to the exhaustion level.
A fighter would likely have the quickest recovery, as he could quaff a potion and use Second Wind on the first turn that he's back up (though Second Wind wouldn't be available if he fell to 0 HP again), allowing him to be at exhaustion level 2 (half speed, but no attack disadvantage) on his second turn after being healed from 0.
... Honestly, I kind of like this idea. Might try it as a house rule.