IME the answer to this always seems to be... give him magic but fictionally disguise it. IMO WotC solved this problem in the best way possible for their game in 5e... everyone has the option to choose magic. If you feel there is a giant discrepancy between magic wielders and non-magic wielders then good news... you can pick a subclass that uses magic for any class. If you don't (or don't care) it's a moot point and you can choose to forego it.
I think I prefer this method to the "magic as skills & abilities" method since I almost feel like that is it's own genre that isn't general D&D (though I'd have no problem with a campaign setting where it is the norm). That genre is exemplified by games like Exalted, Earthdawn, Godbound, Ninja Crusade, Legends of the Wulin and so on. Just my opinion though.
Whereas, for me, "just pick the option that has literal spellcasting or disguised spellcasting, 4head" is not acceptable, at all. I am very glad that such options exist, because they are great for people who like that sort of thing.
But they cannot, not even in principle, satisfy the desire for "person who, by mighty thews and deft hand and skill with weapon alone, slays the terrible dragon." An extremely common, well-supported character archetype in myth and literature. Indeed, frequently the hero of the story. That D&D fails to support this archetype as an equal participant has always confused me.
I do agree however that chosing a wizardy subclass is no solution. Instead look at the rune knight to have a supernatural fighter that does not use spells and is very powerful.
I appreciate that you are trying to extend an olive branch here, but the difference between EK and RK is not relevant to me. They are both magic users. One uses spells, the other uses runes, but they are both magic users. I want to be able to play a Fighter that does not use magic but keeps up with the Wizard or Druid or Cleric that can drop 25 spells a day.
For a seemingly mundane fighter there is no great option I fear. All attampts beside battlwmaster just look inferior to battlemaster...
And Battlemaster itself falls short on multiple fronts. (As Ruin Explorer noted, spellcasting can keep up with the damage of Fighters while also having spells left over to do other things that are simply impossible for the Fighter.)
The Spell slots have a specific mathematical value, in HP terms.
Show me the numbers. Unless and until we can actually critique the numbers, my only response on this is, "Objection: Hearsay," to quote the recent meme.
Because frankly the few references you've made sound like they ludicrously under-value the benefit of spells. E.g. I believe you said
hold person was valued at 3d10, which would be hilarious if it weren't so infuriating: a successful hold person is worth the number of attacks the target cannot make because of the spell, plus the number the PCs can land against the target (because it doesn't get extra saving throws for being attacked), which have advantage if made in melee.
Let's say the target has a 40% chance to make the save (meaning, the spell has a 60% chance to work, comparable to hitting with an attack.) Mathematically, that works out to an expected value of (somewhat surprisingly) exactly 1.5 average enemy turns under the effect. So, you negate on average 1.5× the creature's average damage, and get 1.5× the average damage (accounting for hit rate) of the party, with melee characters factoring in advantage. And that's supposed to be 3d10. Even at level 3, when hold person first comes online, that is obviously an extremely low-ball estimate of the spell's effects. Which would certainly be par for the course of 5e design.
