You are correct. The fighter (and rogue, and to a lesser degree the barbarian) should NOT be mundane. They should be magical.
the fighter, rogue, barb (and tentatively monk)
should be mundane, what they
should not be is ordinary OR explicitly magical/supernatural (barring subclasses who's point is to be such), trained strength, mastered skill, forged steel, all of these are mundane, and all of these should be enough to take down a dragon with in worlds where dragons are allowed to exist.
being mundane does not mean you cannot be extraordinary but being extraordinary does not mean you have to be supernatural.
The fighter should never be "dude with a sword".
this is explicitly the class fantasy that i believe half the people who pick fighter
play it for, to be 'just the guy with the sword and shield or spear, axe or hammer' who through pure mastery of combat keeps up with the wizard throwing around fireballs and hypnotic patterns or the druid who turns into a bear and brings the forest alive to attack.
The fighter should be "dude with supernatural ability who channels it into his sword". He should be born of the Gods, have dragon's blood in his veins or heir to the giant's legacy. He should breathe fire, sheath his weapon in energy, sprout wings and fly, and at 18th level have abilities that start "once per day, when you die..."
half of these sound more like descriptions of lineages than anything to do with fighters, why would anyone with those traits not equally turn to become a cleric or artificer or any other class? 'born of gods: aasimar' 'dragon's blood: dragonborn' 'heir of the giant's legacy: goliath' and i don't know since when breathing fire or sprouting wings is thematically connected to mastery of combat (again, barring a specific dragon themed style/subclass)
you only listed two abilities that i thought sounded fighter-like which i bolded: nigh supernatural skill with weaponry and enhancing their weapon, the 'when you die X happens' is something i can imagine the basis for abilities that any class might have and iirc was something that one of either 3/.5 or 4e
did have for all classes?
the fighter to me should be the guy who can fight with any weapon from blades to bows, boomerangs to barstools, or just bare fists and still take on an entire crowd and win, the guy who can lead pesant farmers for battle and have them be as effective as town soldiers, the guy who can grapple and throw a giant, who can plot out the most advantageous places and strategy for battle and improv on the fly, who could march for three days and three nights straight carrying all their gear and be ready to throw down as well at the end of it as the start, to snipe whichever apple you decide off the tree on the other side of field, who throws greatswords like they're naught but daggers.
A cleric isn't a village priest. A rogue isn't a common pickpocket. A wizard isn't a local scribe. A fighter shouldn't be a town guard of infantryman. Mundanity is for NPCs.
a NPC isn't an inherently different creature to an adventurer, oh we might build them with different mechanics but there's no reason those examples couldn't be true, and adventurers had to be learning somewhere before they became adventurers, there's probably alot more level 1-3 classes among the general populus than DMs bother to put in