Except there is an absolute ton of literary and mythological precedent for fighters being able to do completely amazing feats that fly in the face of realism. Non-wizards being able to do amazing things is a well-established part of the broader fantasy genre. Expecting fighters to be awesome is an integral part of my expectations out of fantasy, mostly because it is heavily shaped by mythology.This is about like admitting that a long infantryman is not the equal of an apache gunship or main battle tank. I fail to be amazed.
To make that infantryman the equal of a gunship or tank requires giving him superhuman powers. If your genre conventions forbid that, then you must look outside the infantryman and let him go to the quartermaster to pickup a stinger or TOW missile.
In D&D terms the fighter needs magic items, or a flying mount, or a team of guys with a catapult. You can do this.
What you cannot do is make a guy with a big knife the equal of a man who commands the elements and the forces of life and death. Or at least, not without turning him into Goku. And you also cannot insist that I play Dragonball RPG when I wanted to play D&D.
Heracles didn't need any fancy magic items to slay nasty monsters and rearrange European geography. Beowulf tore Grendel's arm off with his bare hands. Thor could drink the ocean down a few feet of sea-level and fished up a serpent so big it encircled the world. Cuculain slaughtered an army of men so strong they carried spears as large as pillars by throwing pebbles at them. These heroes of myth didn't need magic in order to be strong and impressive, and were not chained to some silly idea like realism.
People have been trying to argue that this part of mythological influence is somehow foreign to fantasy, but that idea is silly. J.R.R. Tolkien was heavily influenced by this mythological legacy, and it clearly shows in his works. There is nothing mundane or realistic about his elves, or the men of Numenor. Aragorn lived for more than 200 years, after all; that is not something a "mundane" human will do. The mythological influence of Tolkien is even more clearly seen in the Silmarillion.
Fantasy itself evolved out of myth and fairy tale, and in the world of myth and legend, heroes are not constrained by such petty things as realism. On the contrary, heroes are defined by how they defy and exceed human limitation. What makes characters like Heracles so interesting is that they are so much larger than life. In my book, telling a character that they have to be "mundane" is akin to telling them that they can't be a hero at all!
And I for one want my characters, be they fighters, wizards, or thieves, to be heroes.