Something else you should consider is to keep the Death Saves but lose the 'fall prone and unconscious' aspect. It's a very minor mechanical change that i think can dynamically impact combat thematically.
So when they hit 0hp they stay up and when it's their turn they keep fighting as normal but at the end of the turn they make a Death Save, if they fail check it off. Since they are still up and fighting NPCs will keep attacking them and each attack still adds a failure, a crit marks off 2 failures. Get 3 fails and your dead.
So basically it's the exact same system except your on your feet fighting for the whole thing. It seems like the tension will ramp up once you hit zero, but also adds a buffer, and is fictionally keeping with your theme. It will keep the PCs alive longer, but most creatures have multi attacks and hitting PCs isn't very difficult and damage is irrelevant.
If you want to keep the mood unforgiving, you can also make it so that healing doesn't get rid of the Death Save success or failures. You can say that "It takes 1/10min for your Death Save to reset" or a short/long rest, or even that they don't ever reset.
I like the idea of getting reduced to 0hp, then getting attacked twice, then on your turn you know that this could be the very last thing you do so the turn carries weight to it, because at the end of your turn you can roll a fail on your Death Save. Will you attack the creature and kill it and then die with it, will you heal a friend to keep them from the same fate as you, will you dash away desperate to live.