The thing is that there's enough variety between the classes that you propose removing that I don't think that a single hypothetical class can cover them all. It's never existed in any edition (although some Gish-types have) and the balance between spell and sword is so different for the different subclasses that you'd need something very fiddly.I talked about this in an earlier post. If I were in charge, I'd get rid of those redundant subclasses. We don't need an Eldritch Knight (which barely functions mechanically as a gish), a Bladesinger (super restricted in armor/weapon types), and possibly even an Arcane Trickster if there is a base class for the "Arcane Gish" idea. There is still a design space for the Arcane Gish class in D&D 5e, but there also is too much overlap with subclasses that don't fulfill that niche.
And you say the bladesinger is "super restricted". I'd say that's part of why it works thematically. A sword in one hand and spell in the other is iconic. And that even most sword wielding mages don't wear much armour either because they can't wear it because the armour interferes or simply because they are protected by magic. It's an iconic combination - and if it didn't have to do this most wouldn't.
And @Frozen_Heart I can't agree that the duskblade, that capped at 4th level spells, was anything other than very slightly better than the Eldritch Knight. Meanwhile the swordmage was pretty good - and is an entirely different class and approach from the magus. And a duskblade certainly didn't cover the same sort of gish as the bladesinger.