Fighting skill: STR, DEX, both... or none?

In OD&D, Fighters didn't need to rely too much on Strength; it was a (fairly modest) bonus to XP, nothing else. Then, abilities got more and more important (mostly, in other RPGs), until it reachead  peak in 5e. Now, because of bounded accuracy, the fighter's Strength (or Dexterity; see below) is often as important as level when determining how well you can hit your opponent.

Even though HP and number of attacks will still make the higher level character win most of the time, in WotC era D&D (and, to be fair, in much of TSR-era D&D), having a fighter with low Strength is unthinkable.

There is nothing inherently wrong with that idea. In fact, I'm more annoyed with the opposite: the fact that now Strength is the most likely dump-stat for Fighters based on Dexterity, since it does so little aside from combat (which the DEX-Fighter doesn't use), armor (which the DEX-Fighter doesn't use), and encumbrance (which the DEX-Fighter doesn't need, since he is probably wearing no armor).

The purpose is obvious (and laudable): to allow players to create nimble, fast warriors archetypes like Robin Hood or Legolas. The downsides are also obvious (and laughable): strength becomes useless when wielding a longbow!

Conversely, DEX might be a good dump stat for STR Fighters: they cannot use it to gain better defenses when wearing heavy armor (which is silly IMO), or to get a combat bonus. The reason that "Dex as a dump stat" happens less often is that DEX is so useful to so many things: combat with finesse weapons, AC, initiative, acrobatics, lock-picking, driving a chariot, playing stringed instruments and crafting small objects.

This list (from 5e, as written) sounds a bit ridiculous; in the game, it seems like too much cramped in a single ability, and in reality there is no direct link (that I know of) between an ability to play the guitar, dodge a punch and shoot a bow! Even leaving "gamism" and "realism" aside, the fact that this things are tied to a single ability is detrimental to creating archetypal characters that are good with longsword AND longbow, for example, or that can move fast while in armor, can play the lute but not dodge a punch, etc. Why would Robin Hood be as talented as Will Scarlet with a lute (at level 1 anyway), for example?

This is NOT a D&D-only problem; DEX is the uber-stat of many RPGs, that often manage to be a lot WORSE than D&D in this regard by making combat solely dependent on dexterity.

Also, can you think of an archetypal fighter that is a weakling? Even fighters that are known for their speed and accuracy over raw strength (say, Legolas, Robin Hood, Black Widow) would beat an ordinary person in an arm-wrestling contest, in my mind. What about the hulking brute that is an excellent fighter despite having no dexterity at all? That archetype is more common, but is most often used to be defeated by the weaker protagonist. Conan, Fafhrd, the Mountain (ASOIAF), etc, are certainly NOT good examples of clumsy heroes!

No dump stats for this guy!

What is worse, the current system DISCOURAGES "well-rounded" fighters with both good STR and good DEX, which goes against archetype, fiction AND reality!

If STR and DEX affect how you fight your battles, they should BOTH be important; letting STR define damage and DEX affect defense (no matter which kind of armor you're using) sounds good to me.

Now, which of the two should define "to hit" capabilities? Well, since "hitting" an enemy is an abstract thing is D&D, any solution could work.

Maybe both, since strength and dexterity are both obviously useful for a fighter? Again, nothing wrong with that, but it makes the game needlessly more complicated. In fact, one could argue that intelligence is very important to a fighter. Willpower certainly is, and perception too. Fitness, obviously (although this is reflected in HP). And so on. Potentially, any ability can be useful for fighting. IIRC, the Radiance RPG does exactly that, allowing characters to use their main ability for fighting.

Still, fighting is very much its own thing. The best fighter isn't necessarily the strongest, or the fastest. Connor McGregor's (the current UFC Featherweight Champion sentence after beating the (awesome) former champion José Aldo (using plenty of provocations and mind games, one might add - not to take anything from such an impressive victory, but just to show how the mental aspect is important in a fight) sums it up nicely: "He’s powerful and he’s fast, but precision beats power and timing beats speed".

Charisma and willpower are often important for fighters

Dexterity and strength are important, but not as important as how you apply them. If that is the case, why not just let fighting be a separate thing, as it was in OD&D? After all, magic-user didn't get spells of higher levels because of their INT (although they did get extra spells in some versions).

The damage boost is good enough for the strong fighter, the AC bonus is good enough for the nimble fighter, and (although no fighter can completely ignore any of these aspects), if fighting ability is defined mainly by level, an expert fighter can succeed against a faster or stronger foe.

This seems more in tune with reality, archetype, and, what might be even more important, makes the game a bit simpler and more flexible at the same time.