Can't cast somatic spells while dual-wielding,
What do you mean by daul wielding? Are you talking about TWF or just wielding a weapon in each hand? RAW Nothing prevents a character from wielding a weapon in each hand on the same turn he uses a BA spell. RAW - A character that starts a turn with a weapon in each hand can attack (with one weapon or with both if he has extra attack) and then drop (or sheath) one of them and cast a BA spell. If he starts with only one weapon in a hand he can cast the spell and then draw a weapon into his free hand and attack with either or both weapons. So RAW there is nothing in the rules to prevent this. The only thing it prevents is TWF and casting a BA spell and that is based entirely on action economy and has nothing to do with ths somatic components (TWF uses the BA).
If you house rule this essentially you are bringing back bonus actions but just call them something else. You are ruling if you use a TWF "non-bonus action" you can not cast an "instantaneous non-bonus action" spell .... but at the same time you can still use cunning action, rage, bladesong or martial arts, which makes no sense.
Finally, I want to clarify that any character with extra attack can already "dual wield" for free, they just don't get a third attack through TWF. RAW a 5th-level fighter (any 5th level fighter) can attack with a longsword in one hand and a battle axe in another, nothing in the rules prevents that. The only thing the rules address is the extra attack you can get through TWF (if you meet the other limitations). It is the extra attack on top of the normal attacks that uses the bonus action and this is the reason it is a bonus action.
Advantages of TWF:
1. You can go DEX-based, and DEX is the god stat. IMO, if TWF can hit as hard as GW, there's almost no point for STR fighters to even exist.
2. If STR-based, you can open combat with an extra handaxe attack.
TWF can already hit as hard as a a two handed attacker.
levels 1-4: 2d6+8 (15 average) > 2d6+4 (12ave)
levels 5+: 3d6+15 (26 ave) = 4d6+10 (26 ave)
These numbers include the GWF fighting style rerolling 1s and 2s. They do not include criticals.
If he is a Barbarian in rage he actually does more with TWF (after the first turn). Discounting racial abilities and feats, you have to be 11th level fighter, and fighter only before someone with a greatsword will beat someone with 2 shortswords in terms of at-will damage.
I disagree that this is a balance issue. If Blade Song hadn't been a bonus action from publication time, and Shadow Blade popped up in round 1
People on this board already complain that a bladesinger is OP, with it written as is, and said so before shadowblade was even published, so I am pretty certain many would be agaisnt anything making them more powerful.