Basically fans want to high level barbarian to be the hulk.
Making the high level barbarian into the hulk in D&D would be such a drastic mechanical change and shift philosophy that it would not only require an entire addition change It will require convincing the community to agree to some mechanical and visual changes.
Like encumbrance would have to start mattering. You won't be able to skip encumbrance anymore. The formula for encumbrance would have to be not simple anymore.
I don't really follow that last sentence. Playing or GMing a Hulk-like PC doesn't require complicated encumbrance rules.
Your examples: "You might want to: take the good fight to a Demon Prince, wrestle a Kaiju, time travel, battle a Great Old One, invade Hell and shave Asmodeus' moustache."
All the stories I care about have personal stakes. This sounds more like banging action figures together, which is fun on occasion but doesn’t sustain my interest. I suspect most people feel the same, which is why high level characters are more fun to think about than to actually play.
I think high-level D&D-style fantasy works best when
what is personal and
what is "external" are closely integrated. If the lives, hopes etc of the PCs are disconnected from the mythic reality of the (imagined) world, then high level play doesn't offer anything of particular interest.
Aragorn-esque or Arthurian "true kingship" stories are one example; but there can be others too.
In my high-level 4e D&D game, the PCs defeated Orcus - thus realising a crucial ambition of the Raven Queen, who had two exarchs and another powerful follower among the PCs - and also Lolth, just realising the dream of the Drow PC to free the Drow from her tyranny and undo the sundering of the Elves.
In a high-level Rolemaster game that I GMed, one of the PCs was able to rescue the "dead" god that he served from the void. Another PC was able to help resolve various conflicts among the Storm and Sea Lords, thus clearing the way to - and showing his worthiness to - wed the dragon whom he had been courting for many, many levels.
I don't think there are Emotions you can have at 20th level that you cannot have at 12th level. Which means the difference between the two is going to be in terms of spectacle and (collateral) stakes.
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Scale doesn't need to be a hindrance to story.
To me there's one answer that seems to make the most sense: more people actually care about the stories their PCs experience, not the game mechanics.
All the stories a player can experience can be told at any level of character.
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There's nothing one gains by playing high level other than using some game mechanics you've never used before. But anything having to do with a character's personality? Their history? The journeys and quests they go on? Their stories? Can be done at any level.
I think this is a contentious reading of D&D.
There's an alternative reading, which is that certain sorts of "stories" are level-dependent. For instance, meeting a Pit Fiend as an equal, or carving out a kingdom to rule, or travelling places by way of teleportation - on my understanding of D&D (informed primarily by B/X, AD&D and 4e D&D), these are not things that can occur at any level.
When these sorts of events or experiences are intertwined with personal aspirations and/or self-realisation - which to me is the essence of epic/mythic fantasy - then I think high level play can have something to offer.
I think 4e D&D shows how this can be done within a D&D framework.
And my view - tentative, but based on a fair bit of observation of how people talk about their play, and how TSR/WotC present high level play - is that a big problem is that there is a reluctance to allow the sorts of stakes (and associated themes) that are central to epic/mythic fantasy as a part of D&D play. And of course, high level play that is merely a continuation of fetch quests, looting dungeons, and the like
won't offer anything that low level play doesn't.