Fighters are not getting "left out". It's a character class, not a person. You can't hurt its feelings. But you can leave out players. And if every class goes supernatural,....
First of all, it's not if: every class in 5e has at least one sub-class with supernatural powers. There's just 3 classes that also that have some non-supernatural sub-classes.
Because Heracles isn't just a high-level Conan. Heracles is accomplishing superhuman feats of strength
Sure, but super-human needn't mean super-natural. Strangling a snake is not supernatural, for instance. Snakes breath, you constrict them hard enough, they die. Perfectly natural.
It is true that there's a lot of archetypes out there that specifically /don't/ call for supernatural powers. But, where you put the line between supernatural and extraordinary or superhuman is an issue. There's a tendency - most evident in all the martial-hate/fighter-cast-spells propaganda of the edition war - to draw a line even more restrictive than scientific realism, and label anything that crosses it "supernatural." That tendency needs to be resisted, because /forcing it on others does very much leave out players/, while tolerating different visions of what's supernatural or not doesn't leave anyone out, it just asks them to politely let other people build and play characters to concept, and to engage in this game of the imagination, while, imaginatively. That's not too much to ask.
Wrestling with the embodiment of death to save his friends dying wife?
Well, that's dealing with a supernatural challenge. But, there's nothing supernatural about wrestling. Heroes dealt with supernatural challenges using non-supernatural means all the time.
There need to be classes for them too.
To be fair, not-supernatural does not need to imply mechanical inferiority nor lower-level. Low-level can be used /in place of strict mechanical inferiority/, though, if that's what's desired. If the story calls for a doughty halfling accompanying a mythic wizard, the one can be 5th and the other 17th - BA is even meant to help with that, a bit ( the halfling can roll a natural 20 to guess a secret door's password, that has eluded the wizard, for instance).
The sorts of things he accomplishes are exactly the sorts of things a RAW 20th-level fighter (or barbarian) can accomplish
Well, except for the whole big pile of dead bad guys, and the nettling old issue of being high-STR focused w/o heavy armor. But, that's just one vision of Conan, and not the most authentic. Well, that and Conan also went on to become a victorious general, and a king. It'd be a little tough to be a maxed-STR, high-DEX, high-CON, barbarian, /and/ accumulate stats/skills to be a credible thief, pirate, general and king. Not much easier even if you went with Fighter for the extra ASIs.
The mechanical reason I suggested a full class rather than a subclass is that the base fighter class gets four attacks and Action Surge, which limits the amount of power you can put into other features. If you build a new class around a special Strength progression that eventually gets them to 30 Strength the same way you build the rogue around Sneak Attack, I think you can fulfill the mythic fantasy better.
That's a very specific solution, and probably not the only way (the game /could/ more generally allow for higher stats, without making it necessary everyone take them, for instance - via diminishing returns, perhaps), but it's the right general kind of idea. The fighter class has to be changed fundamentally, or obviated - the way it's designed now, it's too hyper-focused on minimal-versatility DPR to /add/ things to or merely tweak things with a sub-class. (It's not a worse design than any other class, just asked to do too much.) Either that, or you could present alternatives to the way it uses extra attack & Action Surge...