How can dnd (or the wider rpg community) qualify and create a game more attuned to this "something else" playstyle?
Easiest method is to (a) play a different system, and/or (b) create modules that support this "something else" playstyle. Using the second option to introduce setting-specific elements and variant rules that serve the genre you desire to cultivate can provide a play experience more aligned with a classic fantasy aesthetic, but still the expectations will headbutt the rules of hit points, spell slots, rest mechanics, and so on.
As an example of how I might structure option (b) in a classic fantasy experience where you're rescuing damsels from dragons.
1. Characters don't die. If they are reduced to 0 hit points, they're defeated and have to return home. The damsel sighs and says, "When will a hero come rescue me?" and you pick a new hero to rescue her from the dragon's lair.
2. Non-combat conflict resolution is huge. Cleverness, trickery, and diplomacy reign supreme over attack rolls and saving throws. The module explicitly provides rules guiding non-violent resolution methods. Gandalf is better to trick the trolls into arguing until sunup than blasting them with a fireball (because that's lame).
3. Resting doesn't involve hours, it involves the players opting to take a rest, which is a few minutes at most. What happens then? There's a countdown. Every rest brings the heroes closer to the moment when the dragon wakes up and flies off with the princess. They have, say, ten ticks before the dragon overcomes them. A short rest is 1 tick, a long rest is 3 ticks. At 10 ticks, it's game over, try again.
D&D does what it does. Trying to square peg a broad fantasy genre into its round hole of D&D is doomed to failure and frustration. If you want to play King Arthur and his loyal knights, you're best served by playing Pendragon. If you want to play Tolkien, play Burning Wheel. If you want to play a semi-crunchy fantasy game that allows for heroics, play Savage Worlds (but I always recommend Savage Worlds).