When it comes to NPC and motivation to killing blow, I assume they do not have knowledge about metagame mechanics like death saving throws. However, particularly vicious enemies will give the fallen character the typical "sword in the gut" for good measure, which leaves them with only a single death saving throw failure left. This adds a lot of suspense and realism without just saying "you're dead, ha ha!". Additionally, I will let people spend their action point/inspiration/etc to auto-stabilize.
But, I'm a narrative DM. I've also been playing and DMing for 33 years now, and my goal is to create a story framework in which the characters grow. Since I rarely have the same group twice, I tend towards long, epic campaigns, with a strong sense of direction and some appropriate level of sandbox thrown in, though always reasonably minimal. I don't expect my characters to die. Sure, there is tension in combat, but my games have always been characterized far less by "what now" than "what next?". My players usually get drawn into the story and want to know what is around the corner, twists and turns and plot pieces. Combat is a tool that forwards that along, but my characters never really expect to die in any permanent fashion in my game, and I think that's okay. I think Lethal campaigns are fine too, though I find no joy in playing them. And not if I'm dying in "epic combat" to a group of gnolls. It takes me a significant amount of time to craft a character that I can play, so re-rolling a level 1 (or even creating a new level appropriate toon) has very little joy for me. I take the same amount of time to craft truly expansive stories for my players to experience, so I try to make sure they can. Tension from my games often comes from the fact that I keep time and events moving while the players do things. It's what I'm best known for. Plots advance while players rest off wounds or track down a higher level cleric to resurrect a fallen comrade. Dominoes fall and the plot changes if the players take too many risks, or not enough.
That doesn't mean I let them survive sheer stupidity. Especially since I often drop lots of hints that an action is very stupid. But my epic heroes are unlikely to ever TPK due to good monster dice rolls.