I'd be somewhat open to a system where the fighter (and to a lesser extent other warrior classes) could gain in power through equipment OR substitutes for it OR a mix of same. That's basically what Achilles is--he's a guy that didn't need magic armor because his "permanent magic skin armor" mixed with normal armor made him more powerful.
What you need to make this work is some kind of rationale and system where the warriors get access to this replacement stuff, while other characters mainly don't (with the rogue branch as a possible minor exception). Fighters are obviously at the head of the line. I suggest as a starting place that it be built as some kind of direct opposition to magical/supernatural abilities. That is, casting spells or having supernatural abilities makes this "equipment replacement" stuff not work as well for you by its very nature. (As an aside, you could also tie that to equipment, optionally, to reinforce why a +5 sword isn't the wizard's best friend.)
Any ideas along those lines? The obvious mechanical, mostly metagame one is that there is some kind of "essense" of destiny, character, fate, etc. that is unlocked as one levels. It is required to gain those magical and supernatural powers. And crucially, also to wield the high level magic items. If you don't have those, you can spend it all on getting dipped into magical rivers, learning impressive tricks from the greatest retired warriors of your age, etc. Of course when you implement that in a class system, it is going to look a lot like "special abilities at high levels". Perhaps it might be a guide to the character of the special abilities, though, and could be more clearly tied into magic equipment.