There’s an issue in D&D 5e where really a PC is kind of untouchable in terms of mechanics once downed and unconscious. It’s seen as ‘unfair’ to directly attack the PC when they are in this state. In some ways this is good – it’s a soft ‘lose’ condition, in that the PC went down and that feels bad so it’s a way of losing without having to have a PC die (particularly useful if resurrection isn’t possible in the campaign). Some other PC stabilizes them or heals them and then it’s gets back to normal.
So they never die…but on the other hand they never die!
Here’s an idea: The PC has a significance rating – the more significant and meaningful things they do and are involved in, the more points the DM subjectively adds to the rating. It can add up to 25. In a way significance rating is a kind of treasure and the DM could suggest various things gain it (and some things might reduce it)
If the PC goes down the DM rolls an unmodified D20 and tries to get equal to or over the significance rating. If the result does so then an available monster will hit the PC once, but not for a crit! Just a regular hit, making the PC fail one death save. That’s all! Now if the PC A: Has to make a second save and B: Rolls a 1, they actually die! >:)
Even if the DM doesn’t equal or beat the significance rating, the rating goes down by 5 points after the roll. So best earn some more points after that!
This sort of thing I would tell the players before the campaign, either deciding whether I’ll negotiate with the players over it or whether I will insist it’s a requirement (which runs the risk of no one playing. But if something is important to you at that time, it might be better to not game at all than game without it)
The point of a significance rating is that sure, you don’t want that really important character to die. But on the other hand, if they are really important, shouldn’t they be earning significance points anyway? A player can feel they can be more secure in their character without feeling the complacency that comes with plot armour. Keep earning those points!