Tolkien is inspired from mythology, but it doesn't have the same irrational tone as real-life myths. LoTR was so popular because it "modernized" myths into pseudo-rational serious fantasy palatable to a modern world in ways that real-life myths are not.
I disagree with your assessment of LotR popularity, and would state that it has remained popular so long precisely because it emphatically did not cater to the modern mindset.
I have views about the cultural politics of LotR, but this thread (and this forum) probably isn't the place.
But if we just confine ourselves to the "dream logic" issue, LotR displays it. One example: the Hobbits live in an essentially autarkic backwater. Yet they have a material standard of living comparable to that which emerged in Britain once it became a centre of world commerce and industrial production. How is this possible? Answer: in the real world, it's not. In LotR, though, economics and sociology unfold according to a mythic "dream logic".
A second example: what actually takes place in Gandalf's 14-day fight with the Balrog? How many blows are struck? Do they all miss, or are the two hewed apart again and again, or do they regenerate? The text has no answer, and doesn't present the desire for an answer as even salient. That 14 days unfolds according to dream logic.
A third example: Aragorn's 80 year career as a tracker, mercenary, soldier, ranger, etc that culminates in him rescuing the Hobbits at Bree. What does he do for 80 years? How is he not notorious and famous? How can the Bree-folk see him and think of him merely as "Strider, that sinister ranger"? It's dream logic.
Then there's all the "divine power of kingship" stuff - the hands of the king are the hands of the healer, the crowing of the king is a guarantee of piece and prosperity, etc. This stuff isn't about Newtonian, or even post-relativistic, cause-and-effect. It's the logic of dreams, of symbols, of tropes that resonate because of their weight in the literary culture that Tolkien is borrowing from.
Even the long lists of kings and stewards in the Appendices have a dream logic to them. What were they all doing for the thousands of years between the arrival of the Numenoreans in the second age, and the restoration of Aragorn to the throne? How come no social change seems to have happened in all that time? These aren't historical texts - they're a literary analogue to a declaration of (conjecture, mythic) ancestry made by a pre-modern war leader, with the same inherent logic (ie a dream logic) as the discussion in the English classics (Geoffrey of Monmouth, maybe?) of Arthur having conquered Rome.
Giants are impossively massive because couldn't absorb enough oxygen to survive.
Well, there is no oxygen in most fantasy worlds, including standard D&D. No nitrogen. Just the Element Air. (Dinosaurs could grow to gigantic sizes because the air was so oxygen-rich during that era; the Element Air is similarly different in properties.) The Element Air supports all living creates of all sizes.
Done. Giants are not impossible after all. Fighters mundanely chopping mountains can still be impossible.
Scientifically, dragons would be too massive to fly. We can say that all creatures with wings can mundanely have lift in Element Air.
Done. Dragon flight is not impossible after all.
There is nothing in Tolkien like this - no attempt, for example, to explain the physical process of turning the world into a sphere (and what would a non-spherical world even be?) while preserving, for some sailors, the "straight road" to the West.
And I'm not a big fan of this sort of stuff in fantasy gaming. To me, it is like the pseudo-science in the Draconomicon or the midichlorians in Star Wars - a genre-breaking attempt to impose hard science fiction norms of storytelling onto fiction and tropes where those norms have no work to do, and don't belong.
What bugs me is what I surmise is the goal of cherry-picking and choosing mythical elements that make my fighter more mechanically powerful while ignoring the inconveniently accompanying mythical context, such as talking animals or whatever that don't jive with the D&D genre
I heard a player at another table once, in all seriousness, say that talking animals weren't acceptable to him because he wasn't playing (colorful language omitted) "Watership Down."
I also think talking animals are a special case, and I somewhat share the (colourful language omitted) "Watership Down" response. But only somewhat. The Rolemaster campaign before my current 4e one had as one PC a fox spirit in human form (inspired most immediately by the movie Green Snake, I think). And sphinxes, dragons, naga etc are all, in some sense, talking animals.
More generally, though, dispensing with some elements of mythical context in the interests of playability, getting rid of mediaeval stuff that justs looks silly, etc seems fine to me. It's the need for ingame causal explanation as a barrier to entry that I don't really get - especially because, in my view, none is actually being provided for wizards, warlocks, sorcerers etc.
Fight your way into the underworld and rescue them.
That's one helluva class ability!
And some 4e martial PCs have abilities pretty close to it. A couple of examples from the Dark Wander epic destiny in Martial Power (prerequisite either ranger or rogue):
Dark Road (24th level): You can walk to any destination you desire in a single, uninterrupted 24-hour period of walking. No matter how distant the location, or how many planes separate you from it, you reach the destination 24 hours after you begin, finding shortcuts, portals, or other modes of transport previously unknown to you. You do not require any rest, food, or water during this travel, except to recharge powers and regain healing surges. During your journey, you are safe from hazards, attacks, and other dangers. . .
You can choose to be accompanied by a number of characters equal to 5 + your Wisdom modifier, all of whom share the benefit of this class feature.
Long Walk Back (30th level): If you die and are not returned to life within 12 hours, your body and possessions disappear. Twelve hours after that—24 hours after your death—you arrive, equipped as you were when you died, having just walked back from
wherever it is you and your DM decided you awoke after you were slain. . .
You can choose to arrive at the place of your death, at the location of any of your allies, or at any location you consider home.
The point of quoting this is to show that there are actual mechanical precedents in the existing game for mythical fighters. There's no reason why we can't have more of this stuff, with approriate variety in flavour, degree of detail in action resolution, etc.
Fighters (regardless of their archetype) are direct, overt force applied to a problem.
The problem with the DnD Fighter is finding a way to make direct, overt force be compatible with genre expectations, mechanically interesting, relevant and viable when compared to his magical counterparts in mid and high level play
while not offending the sensibilities of the sub-section of the player base that wants to keep his deployable resources in-line with their expectations of real world musculoskeletal kinesiology/physics/bio-physics (while he fights dragons -flight - and giant spiders and other creatures with exoskeleton - exoskeleton size - that are unbounded by his design limitations).
A nice statement of the problem.