@Talltomwright I've actually implemented something that I think fits your goals but goes in a bit different direction. It relies on just one rule change.
Instead of falling unconscious at 0 HP, they are Slowed (as per the spell). This condition continues for 1 minute AFTER regaining HP.
Everything else works like normal, auto death save fails when taking damage at 0 HP (but no auto crit 2x fails from adjacent melee because you're not incapacitated), death saves reset once you have more than 0 HP, you still start your turn by rolling a death save, etc.
So dropping to 0 both gives the adventurers a chance to do something heroic in their last moments or get away, heal, etc, yet because the Slow doesn't end after regaining HP, they are still at a disadvantage for the rest of the fight, but not the next encounter.
Maybe a Constitution save to regain consciousness but still be in the "stunned" condition (no action, reaction, or move; speech is faltering; Strength and Dexterity saves fail; and attackers gain advantage) until restoring to 1 hitpoint.
When you are reduced to 0 hp you may choose to attempt to use Concentration to stay Conscious. If you wish to to do this make a Constitution saving throw, where the DC equals 10 or half the damage which took you to 0hp, whichever number is higher. You automatically lose Concentration on any other spells or affects. You can voluntarily end your concentration and go unconscious at any time.
Staying conscious places your body under great strain. At the beginning of every round in which you are conscious at 0hp you take one level of exhaustion and make a Death Save at disadvantage. (Normal rules for rolling a 1 or 20 apply.)
Staying conscious reopens your wounds. Whilst concentrating on remaining conscious you cannot be stabilized by any means (for example, by a medicine check, healer’s kit or Spare the Dying) except by being bought back to 1 hp.
Only successful Death Saves whilst Unconscious count towards the three needed to become stable.
We removed death saves, though. You will only die if you accumulate enough levels of exhaustion by failing your save each round or by taking damage (see below). We didn't like tracking death saves and levels of exhaustion.
Taking further damage places you in greater peril. If you take further damage whilst conscious you must make concentration checks in addition to taking failed death saves, as per the normal rules for concentration and death saves.
PCs can certainly stay conscious, but from my experience with levels of exhaustion you have to worry about how effective they will be as the penalties pile on. So, we allow additional levels of exhaustion based on CON modifier to mitigate this. You'll have to see how viable this is when the penalties poor on.
It is definitely dangerous, taking damage equals a failed death save anyway, so that is something...
Definitely won't break the game.
Exhaustion is a good short-term negative consequence, IMO.
If you don't mind check for death saves and tracking exhaustion, it probably isn't too much. A couple sessions will tell you fairly quickly if it is too much or not enough.
I've done something vaguely similar a few times. Just let the PCs stay awake at 0 hp. Death saves aren't rolled. But they are marked for damage, like normal. Crits mark two death saves, any other damage marks one. This prevents pop-up healing (something I really despise) and it also lets the players keep actually playing the game, instead of being unconscious on the ground until healed. This messes with various things that are meant to stabilize a dying character, but it has seemed to work all the times I tried it. The only question is what removes the marked death saves. I mostly stick with magical healing or long rests. Still fiddling with that though.
Just to say a quick big thank you for all the suggestions and feedback! There's a lot of very different ways of approaching this, evidently, so I'm going to gently play test some of them in upcoming sessions and I'll feedback how they go!