I think that's a viable solution for one's own game, but it doesn't really work in the larger sense of the game as designed.
Magic items can be problematic because they aren't typically baked into the class.
If a DM changes magic item distribution away from the norm (running a low magic campaign), it would impact class balance moreso than now. I recall having a conversation with a DM back in the day. He was a 2e DM who had started running 3e. However, he was approaching 3e magic item distribution from a 2e perspective (significantly less magic items) and was explaining that his players were complaining that they felt his game was too hard. I explained to him that 3e presumes a certain amount of magic for each character, as indicated in the WBL table. To which he responded that he didn't want to give out that much magic... IMO, a viable solution should work reasonably well regardless of whether I'm running a no magic campaign, a Monty Haul campaign, or something in between.
Additionally, magic item distribution can be a bit wonky in some groups. In my newbie group, the wizard player is kind of the leader of those guys even outside of game. He's generally a good guy, but he has kind of a forceful personality and can be a bit greedy in game. Several times he's tried to call dibs on powerful magic swords, and I've had to remind everyone at the table "You realize your wizard isn't proficient with that?". I'll grant that isn't going to be the norm, but even so, if the barbarian convinces the fighter to let them take a magic item meant for the fighter, it shouldn't dramatically impact the power disparity between them. If martials are balanced using magic items, I think it would result in a noticable disparity.
That said, you could just bake magic items into the class. That would be a viable solution, though you'd need to have a way for those items to be recovered if they are lost/stolen/destroyed (which is entirely feasible).
Personally, I'd love to see martials grow to be mythologized at higher levels, becoming legends like Beowulf. I'd love to see a high level fighter be able to rip a giant beast's arm off WITHOUT the aid of magic items. Or a barbarian whose blade has soaked in the blood of so many slain enemies that it becomes a dangerously powerful magic weapon. Or a rogue who is such an unparalleled thief that he can steal intangibles, like the love someone holds for another.