RE: the narrative.
In any fiction, death doesn't occur because the author throws dice or darts at a dartboard (well, I'm not so sure about George R. R. Martin...). Death occurs because the author decides "this is the time a character dies" which happens when they feel it is appropriate.
In simplest terms, when a D&D (or any ttrpg) character avoids death, narratively, it wasn't their time yet. They got lucky, they pulled through, adrenaline allowed them to narrowly avert death- you see this stuff all the time in novels. What some people are having a problem with is that the game itself doesn't tell you what the reason is. Gary Gygax infamously muttered some stuff about divine protection, luck, and morale, and while all of those could be factors, the game engine doesn't tell us why the gods care, what makes the characters so lucky, etc. etc., or when an attack fails to kill a character due to good fortune, skill, or just plain guts. It's left up to the players and the DM to figure it out.
Again, Micah Sweet often points to the example of being poisoned by a knife blade or a snake. This is a scenario where you definitely were struck, did take at least some damage, otherwise you wouldn't be making saving throws. Now, it's entirely possible that a mere nick or graze is enough to be exposed to the poison, but hit point systems don't really define what a hit point of damage means- there's no system for fractions of hit points. You nick yourself shaving and bleed a bit. That's a wound, right? Does that mean you took a hit point of damage? Probably not, but you certainly took some amount of damage!
People talk about hit point bloat all the time, how ttrpg heroes become far too tough and resilient to be afraid of most forms of injury. Most of this has less to do with hit points, and more to do with damage scaling. If a game designer decides a dagger inflicts 1d4 damage, and that the goblin stabbing you has a +2 damage modifier, then at most you can take 6 damage from a normal attack.
Now if the game designer also says a normal person has 4.5 hit points on average, that means on average, a goblin stabbing you with a dagger is instantly fatal to normal people! Who needs swords or greataxes in such a world, when the vast majority of people (including goblins!) die from a dagger wound!
Quickly, however, the game realizes that a system where everyone goes down from a single hit isn't particularly viable, so some form of damage mitigation system is put into place. HIt points being among the simplest versions of this. So suddenly character A can survive two dagger hits, and monster B can survive 4. But quickly, this gets out of hand due to the needs of the game- if a party of four heroes can kill a monster in four attacks, that's entirely too quick, it doesn't feel challenging at all!
So the monsters get more hit points to survive longer. But that allows them to deliver more damage in return, so we have to give the heroes more hit points to make up for this, and this quickly gets out of hand. And then the designers, realizing how silly all this looks, try to get cute and dream up other forms of damage mitigation to stack on top of hit points, be it damage reduction, "phantom" hit points, or regeneration.
I've heard DM's complain about falling damage, that characters should die from falling off cliffs. However, people have fallen from ridiculous heights in the real world and not died, there's nothing magical about that, it's physics in action- what did you land on, how did you fall, and so on. Vesna Vulovic was just a stewardess, not an action superhero, and she fell 33,000 feet and survived with less than 20 hit points to her name (the minimum D&D would inflict to you)!
Of course, it's not the hit points that's the problem, I think. It's that the character doesn't suffer the massive, painful injuries afterwards. As long as they have 1 hit point to their name, they can stand up and walk away. "Surely," some say, "they should suffer permanent injuries, levels of fatigue, and other horrible consequences for daring to survive!". But why though?
Because it "doesn't make sense"? I mean, how much fun is it, really, as a game, if you tell the player "congratulations, you didn't die. Btw, you're going to suffer life-altering injuries that will take months of recovery and you might have to retire from adventuring."
How is that different from death? Your character can no longer play, and you'll likely have to make a new one to keep playing the game!
As to players who act like they are invincible, there's a very simple solution. Their characters should have no idea what hit points are, or how many they have- it's a game mechanic, after all.
So don't let players know how many hit points they have! Roll hit points at level up in secret, keep the totals to yourself, and if needed, houserule any mechanics that would give someone information about how many hit points someone has (like the Battle Master's level 7 feature).
I guarantee they'll become much less likely to engage in reckless actions of any kind pretty quickly!