That's a interesting system idea. The book of nine swords does it to some small extent (not nearly as much as you seem to be suggesting). And it reminds me a lot of the game Nobilis.
Although of course they would be a lot stronger then almost any greek heroes.
That's why I said "based", not "literally a clone". Greek heroes were not very high-level heroes, but they generally have a much better versatility than Fighters. Taking that idea to the logical extension would give us something that wouldn't suck so bad as Fighters do in 3e.
The only supposed strength of the Fighter in 3e D&D is more feats, which are supposed to give the Fighter some versatility, and it would be okay if these feats were actually "feats of strength/cunning/agility", but instead like half of them are "+1 to some rolls in a specific condition" and most of them have other feats as prerequisites, so, unlike a Wizard, you can't mix-and-match your available array of abilities, but instead are forced into one of the several archetypes, unless you want to become weaker than a summoned creature. 5e seems to have fixed that a little bit, but they're still very low-powered compared to a Wizard's spell list.
And in order to make a game fun for everyone at every level, a level-appropriate challenge should be beatable/passable by every class in the game. One class may do that easier than another, or maybe with less side-effects (like if Fighter throws the world in order to move between dimensions, that's gonna cause some earthquakes to happen), but the core idea should be here. Otherwise we get situations where some groups simply can't pass a level-appropriate challenge or where half of the group cannot contribute to the task at hand in any way.
Given that D&D is generally a high-magic setting and it's magic part is the most well fleshed-out (I'm pretty sure most people still play D&D because of the obscenely huge spell list), it would be easier to rebalance other classes to the Wizard tier than it would be to down-power the Wizard to the other classes tier - and there would be more actual game content left in.