Did someone seriously just suggest using an eldritch knight as a mythical fighter? Does that person have no grasp of the argument here at all? Are they trolling? I can't believe anyone would be thick headed enough to suggest such a thing.
For starters, mythical martial heroes do not cast spells. That is what makes them martial heroes and not spellcasters. Beowulf, Lancelot, Gilgamesh, Cucuchlain, and Hercules never once stooped down to the level of "jazz hands and jibber jabber" to defeat their foes. They accomplished tasks using strength, cunning, and endurance. Sure they exhibited superhuman levels of these qualities, but they were never supernatural or magical.
They never had ammo/spell slots, arbitrary daily limits, memorization, or obviously magical effects. Their capabilities all fell within the realm of what one would expect a high level warrior to be capable of. No turning invisible, shooting balls of fire, and flying. Those are superpowers (or wizard spells) not martial exploits.
The EK is anything but the mythic fighter people are asking for. I can't honestly believe someone seriously suggested that as a solution...