I still feel like "the fighter can take non-combat feats" isn't a fair comparison. The wizard can prep level+Int mod spells.
Nod. The wizard is more flexible, sure - it's more flexible than it's ever been, that way, really. Even compared to the notorious Tier 1 Wizard of 3.5, the neo-Vancian 5e Wizard's spontaneous casting gives him superior flexibility within an adventuring day. Comparing that to the fighter is hardly fair.
As long as they prepare a few offensive spells, their combat ability is covered, meaning they can prepare a bunch of social or exploration pillared spells with the rest. If the day ends up being all about combat, then they use their spell slots on combat. If the day ends up being mostly sneaking through a fortress, then they use a lot of spells on invisibility, charm person, and fly.
And, if they actually manage to prepare a slate of spells so bad that none of them are ever useful the end the day with all their slots wasted. (OK, that's a bizarrely improbable corner case, I admit.)
If the fighter picks up languages or extra skills or those nifty new expertise skill feats, they still have their high combat potency floor but they have chosen to not pick a permanent increase in their combat power
Doesn't really matter that much to the point you've made, above. Assuming a fighter puts his two net bonus ASIs towards some extra skills or other non-combat-meaningful feat, he's as good at combat (with weapons) as any Ranger or Paladin with the same fighting style and the same ASI & feat choices. Which is surely good enough. But whatever those two feats are, he doesn't get to change them at the end of the day, let alone the middle of the adventure, and that's a profound lack of flexibility compared to the case you've made for the wizard, above.
The rogue, who has nearly the same combat potential as a fighter, especially against a single target where uncanny dodge really shines, has abilities like expertise to enhance their ability to do things out of combat. Most fighters have nothing of this nature. And, again, the primary casters take the cake on this one.
Nod. I'm sure y'all've heard this before, but you could combine the Fighter & Rogue into one class, taking the best of each and cutting nothing, and it still wouldn't be overshadowing anyone...
What I think is needed is an expanded skill system. More things to do with skills could help the noncasters to do more.
Depending on how you expanded it, it could also just serve to broaden their incompetencies. If you just added more skills and more detail to the skill system, for instance, you'd just have more options for the limited set of proficiencies & ASIs to chase after.
Now, maybe more elaborate skill uses (group checks are as inclusive as it gets in 5e) in exploration & interaction could help - like 'complex' skill checks from 3e or 'skill challenges' from 4e?
Yes, the casters have skills too, but I liked how 3E all but skill taxed the casters with knowledge, spell craft, and concentration. I'd go back to something like that and then increase skill proficiencies.
Concentration was very useful, though, it negated long-standing restrictions on and risks of spellcasting. If 5e casters were 'taxed' with a concentration skill that let them negate the Concentration mechanic, they'd be coming out far ahead. And Knowledge skills are already there, and already useful. And Spellcraft would presumably be useful any time you encounters spellcasters, which, given that every class uses spells to some degree, and that casters are among the most challenging potential opponents, couldn't help but be pretty useful, too.
But, yeah, by all means, throw a few more proficiencies to the handful of non-magic-using sub-classes. Couldn't hurt.