Well people seem to be focused on the fighter and changing it.
In this thread that'd be more nearly on topic (it seems to be more an 'acknowledging the issue' than a 'brainstorming a solution' thread.
I rarely see posters calling for a new class with these superhuman/supernatural abilities.
No, you wouldn't, because we have plenty of classes with abilities that are both superhuman and supernatural.
What we saw so much of that it was exiled to a separate forum, was demand for a class that would have had, at least at it's core, extraordinary, even superhuman, but /not/ magical or otherwise supernatural abilities.
But the community's pervasive double-standard fuels an overwhelming, knee-jerk, virulent reaction to the perfectly reasonable request for the reprisal of a class featured in a past edition's PH. It really is very hard to miss or understate either the desire for the game to emulate more such archetypes from genre, or the double-standard's resistance to accepting such a thing.
Spells and magic are not the same thing...
All spells are magic, not all magic is in the form of spells, yes. The EK is a fighter who casts spells. The Champion and BM are fighters who do not possess magical abilities from their class, at all. There are other forms of magic, but they don't enter into the comparison.
You said so. You came right and said the fighter shouldn't be able to leap a battlement unless he was an EK casting a jump spell. That's exactly the double-standard I'm talking about.
non-magical classes can already do stuff that's superhuman in nature
Yep, a bit, like I said, their side of the standard applies realism /unevenly/. Can you survive a fall from a great height every time without a broken bone that would slow you down? Well, thanks to hp abstraction, yes. Is that super-human, well, yeah, it's hard to avoid seeing it that way. You could lean hard on the 'luck' aspect of hps to argue against it, but falling damage is a part of the rules that frays quite easily, that way. Is there any corresponding ability to leap to a great height? Nope.
all D&D has done is separate it and say this superhuman stuff is the realm of magic
And that's the double-standard.
As for fantasy hero... does it have a default setting?
In theory, some lame thing at some point, IIRC, but, no it's a generic system. That involves very different design choices than a genre-specific or even multi-genre system, let alone a licensed one. That was the point, D&D is /not/ that kind of system, that's why you can't just blithely re-skin a spell as a superhuman ability.
Because as much as we want to claim D&D is generic and should do all fantasy it's not. It is it's own meta-genre
Yeah, that's a whole 'nuther double-standard.

OK, more of a tautology, really: "D&D models the fantasy genre perfectly! ...because, erm, it defines it's own fantasy sub-genre in which everything just works the way it always has in D&D... yeah."