Dual-wielding H.Xbow isn't supported in the system anyway. You only get to loose one bolt with the primary (as no hand free to reload) and then one bonus bolt with the secondary. Also, the situation envisioned is that with your melee weapon, you give foes a reason to keep back, letting you get a clean shot. I'm not seeing why another ranged attack should be supposed to help with that.
Crossbow Expert in the PHB makes dual-wielding crossbows supported; it is not otherwise. This is cool. This is also a big part of the feat, maybe even the most clear purpose for its existence. Why have the feat at all if it isn't about dual-wielding hand crossbows in melee?
This is why we need to adjust it to carve out that role and similar, while blocking OP combinations.
The reasoning of making foes keep back may not be as direct, but at the same time, this is not the point of melee dual-wielding, for in that case the second strike would be at disadvantage due to the keep-backing. I like your reasoning for RP, but it doesn't apply across the board.
Without further analysis we can make an odds-on guess that any caster with Fireball on their list is better than all martials, at least so far as damage-dealing is concerned.
This is true, but Fireball has lots of collateral damage for all casters except Evoker Wizards and Sorcerers. Wiping out your allies is a pretty big drawback. There are single-target spells, of course, that could be considered.
For me, the ideal balance is where casters are best at X, ranged is best at Y, melee is best at Z and so on. We can make everyone best... just at different things.
Caster overshadow has always been the footprints in the butter of D&D: everyone politely ignores them... explodes into a rant... and then goes back to politely ignoring them.
Great balance in RPG broadens great strategies, letting more character concepts shine. I don't think casters need to be a problem at all, and I don't think a feat (or two) could easily address the issue.
To me I think of playing a martial as playing Batman. Batman is in a world of superheroes with "magical" powers, but he is just an above average guy with lots of great tools. Lots of the fun of Batman is that he is trying to be as good as others with superpowers and shut down incredible foes without that super giftedness. (Although, he may have a super mind.)
In D&D, martials are "ordinary" characters in a world of magic who bravely go toe-to-toe with magic and sometimes bend small bits of it to their benefit. Magic is powerful stuff, but the best martial discovers ways to go on without and against it.
This is just one way to view things, of course.