You can find just as many examples of "heroes who use whatever weapon they come across" all the way from John Carter to John Wick, as you can "heroes who use special gear".
Greek Mythology gives us examples of both- Odysseus uses his wits far more often than any specialized tool, but Perseus basically has nothing but divine gifts. Hercules straddles the line, since he kills many things with just godly might, but also picks up the skin of the Nemean Lion and Hydra's Blood Poison.
At the end of the day, neither approach invalidates the others, it has to do with your abilities and the threats you face. No one is saying a hero can't be awesome when armed with the jawbone of a donkey. But some heroes face challenges more dangerous than even 1,000 Philistines.
In this case, they need a force multiplier of some kind, either a divine bloodline, secret techniques gleaned after training from Hell, or, as D&D popularized from the very beginning, magical swag.
It doesn't matter which approach you take, as long as you acknowledge that the game is built to progress beyond low fantasy as levels improve, and everyone needs some kind of special edge to go beyond mortal limits. Either use magic items, or, at set points, grant players Epic Boons to replicate their benefits.
Or design all classes to progress as the game does, starting as mortal heroes, and eventually becoming demigods. It strikes me as odd that D&D used to do these things quite well. Regular D&D eventually leads to Immortal tier play. 4e noted the distinction between Heroic, Paragon, and Epic play, and gave players abilities to match. Several editions took a stab at epic level play as well, from the madness of 100th level characters in H4, The Throne of Bloodstone, to special high level campaigning supplements.
But somehow now this sort of thing doesn't "feel like D&D" to some people, which is truly bizarre. There's this section of the fanbase who want to focus on low-level play, but rather than say "this game will not advance beyond level 6", they want to go all the way to the highest levels with nothing but larger numbers and relegate any ability beyond what a "guy at the gym" cannot do to explicitly mystical classes.
Which just gives those classes a larger slice of the pie.
Also, when it comes to not wanting heroes to be "fantastic", I just want to point out that just about any race other than Human is automatically "fantastic". Why should an Elven, Warforged, or Aasimar Fighter be unable to perform fantastic feats, exactly? Even Legolas has preternatural abilities on full display in the Lord of the Rings movies, and he's a wood elf with apparently no special powers of magic or healing! What a slacker!
