I care about what happens to Alfonso Derecho Izquierda Quijano, the Kobold Paladin who's in love with a dragon. I want him to have an arc, I want him to grow as a person, and eventually I want his story to end - either in death on the battlefield, or having achieved his dream of wedding a dragon, or coming to the realization that he wasn't chosen by a dragon but that it simply coughed up some armor from a guy it had eaten while flying overhead.
Having said that, I also want to play some games where characters are disposable, including Paranoia. That's fun too!
It's moved over the years from the meat grinders of the early editions to the much more character focused ones now, but some things haven't really kept up.
The big one being character death becoming trivial midway through, and the combat kinda being centered around that. Death doesn't mean much when you just keep reviving them, and it's left up to the DM to figure out how to balance that (since the 5e economy is a mess).
5e long ago should've adopted a downed/injured system like much of the competition, with death being much more rare. Then again there's about 100 other things they should've done, and it's pretty clear they won't, so meh. Guess it just sits on the DM's
If a player really wants their character, I'd offer them the chance to do a spin-off campaign where they're in an afterlife, but if someone dies, that's it.
If you emphasize the "role playing" part, you become a character within a larger play, perhaps even the main character. So of course you don't want to die and roll a new character, just as it would be strange for a character to die in a play, but then the actor come back as a new character in the same play halfway through. The fighting and action become just flavor around playing the role, not necessarily the point in and of itself.
If you emphasize the "game" part of this, the former sounds a bit strange. What's the point of a game where you can't lose? For some, what's the point of a game where you're not fighting much? Of course you die, this is just a paper and pencil version of Rogue (the game, not the class).
And of course there's shades in the middle. 5E's strength is also its biggest weakness. You can do any of these things, however none of them are as good as a system that specializes in those. It's a swiss army knife, and a lot of people have gotten into it because of its popularity and versatility. Which ends up having a feedback effect: people play 5E because other people play 5E. And the people who want to learn a new rules system are exceedingly rare in my experience outside of the die hards.
This is one of the things that I don't like about Critical Role. Heavy, heavy on the role-playing aspects, but you as a listener (and, one assumes, the other characters) don't know any of the characters' backstory except what's revealed bit by bit. It's more than a little challenging to follow, especially if (like me) you have podcasts on in the background while driving, etc. I'm not ignoring it, but my mind isn't always 100% on following the details.
They're very good actors, and it's a great show if that's what you are looking for. It's just not what I want.
I would never run a D&D game but instead I would run Paranoia or Call of Cthulhu or Toon. All of those games are fast paced, the first two are definitely unfair against the player so that a lot of problems of "game balance" just don't happen. If I found some people who wanted to play Paranoia right now I could make up enough random dangerous events to deplete their clones before they ever get to the briefing rooms. (A folder full of prerolls is a big help of course) And in CoC it doesn't matter how much money you make because it can't buy you sanity and there isn't anything you can buy that is going to "break" the game.