Umm, if you want fighters that are absolutely, 100% non-supernatural in nature, why are you accepting the 5e fighter?
Two of the sub-classes are non-supernatural. Superhuman, at high level, in the sense that every D&D character is - able to survive falls from great heights with complete certainty, for the cliche instance - and also in the token sense of cribbing a tiny fraction of the dramatic licence used on such heroes in the form of Action Surge and...
I have non-magical healing at 1st level.
'Second Wind,' yes. And it's a fair amount at first level, though it becomes more trivial at higher levels. It's also been decried as 'trollish regeneration,' in the afore-mentioned double-standard's uneven application of reality-isn't-real "realism..."
...by the same token:
And yet, if i had a dollar for every time I've heard somebody gripe about how warlord healing "makes no sense"...lol
That's another example of how uneven and reality-isn't-real the double standard can in its application of realism. It's fine, apparently, for wounds to carry no penalties - indeed, to not be wounds, at all, for the most part, but to be primarily non-physical factors, skill, morale, luck, endurance, etc that get worn-down in the course of battle - but it's intolerable for anyone else to give any of those factors a boost. Unless, of course, they do so with magic, then it's the double-standard's free pass.
... D&D fighters get even more gonzo with what they can actually do at higher levels than even the source material.
Not so much - though, of course, it depends on the source. There's gritty 'low fantasy' out there that D&D hps, abstract and heavy-handed a mechanic as they can be, might seem to substantially out-do. You don't see a D&D character nursing a wound for weeks, or even at all, wounds don't slow any of them down, ever - they're just lost in the hp abstraction. That's not fighters being gonzo, that's D&D unevenly applying it's bizarre, reality-isn't-real brand of 'realism.'
Yes... it borrowed individual elements and threw them in a mixture... that in no way equates to being able to emulate any genre of fantasy...
Never implied that it did - I suppose it might have done, I suppose, if the design and goals had been very different, but that's not the point. The point is D&D extensively mined the whole of the broader fantasy genre, portions of the science-fiction genre, and myth/legend/folklore from around the world - for monsters, magic items and spells, and then, un-satisfied with that, made up more of its own. It left virtually all the prodigious feats of the heroes of those sources behind. Gygax has apparently said that he expected players to gravitate towards more interesting magic-users as they progressed in skill, so perhaps that's understandable. But, whether that established the double-standard that pervades the community, or whether the preference was one he recognized early, is, I suppose a chicken/egg question. The community was being influenced by and pushing at the game's implementation from the earliest days in EGG's basement, to 'zines and in "Out on a Limb," to BBSs and the internet, today, so could go either way...
Serious question... lWhat feats can higher level D&D fighters achieve that Conan, Fafhrd and the Gray Mouser, Moonglum and other warriors of sword and sorcery can't?
Mostly the same ones as every other high level characters, as artifacts of the abstract mostly-non-physical-'damage' hp system.
But where is that line drawn and why should these be the base abilities of the fighter class as opposed to a subclass that you can opt into?
Sub-class abilities would be fine. A whole 'nuther class would be even better, as the fighter's design space is pretty heavily locked into the DPR 'tank' mode and "must be simple" mandate.
Already we have the argument that there's nothing supernatural about a warrior who can leap over battlements because everybody can jump We even have a fighter who can do this via the jump spell or levitate spell or fly spell if he wanted to... the Eldritch Knight...
Yep, that's the double-standard, again. Wave the 'it's magic' wand, and *poof* all objections to the prodigious leap disappear.
but because his abilities stem from spells he doesn't count?? Why not?
Because leaping over a battlement because you're an incredibly strong/skilled/determined hero in a fantastic world has nothing to do with memorizing spells and mutilating grasshoppers. Because the game /does/ draw that very stark line between spellcasting and other abilities, building the 'fluff' of casting into mechanics, themselves. 5e, in particular, goes out of its way to mechanically model how you do things, as well as just resolving what you can do. It wasn't one of the drums Mearls was beating the loudest in talking up Next/5e, but one of the mandates it had was to make magic "feel really magical" again (not coincidentally, that meant conforming to the community's double-standard more closely than it had in the prior edition), in service to that, casting was returned to a more mechanically distinct (though, ironically, even less exclusive - every class uses spells and almost every class actually casts them) sub-system.
There are other games out there that re-skin much more readily. If you were playing Fantasy Hero, for instance, you could give your 'fighter' the ability to leap a prodigious distance by paying the points for it, and another player could give their 'wizard' a spell that consumes materials, takes time, concentration, incantations & gestures to cast, and gives you a comparable ability for a few jumps, just like the D&D spell. D&D has rarely gotten to close to that kind of design - even at it's closest, you couldn't re-skin away 'source' keywords, for instance...