They attempted it in 4e with skill challenges. Everyone joined in and you need to roll a certain number of "hits" in order to succeed in the challenge.
It kinda worked, but also didn't.
It wasn't a functional system, at first, but it was fixed fairly quickly, so it certainly 'worked' at some point. With only a few years of to develop, though, it had a long way to go to catch up with the depth of D&D combat & magic. 5e could take up and further develop something along those lines.
I would denote such a character you describe as being very little benefit to combat. Do you disagree?
Of course not.
And if not, then why can you not also agree that the fighter is of very little benefit in out of combat situations?
Sure.
I think the contention is that 'very little benefit' is still 'contributing' and therefor nothing to complain about.
So, the hypothetical caster who couldn't know combat spells and got by contributing in in combat via inferior weapon attacks, the Help action, and improvisation would, likewise, have no more to complain about than the non-hypothetical fighter.
It's not a situation where a fighter just has lower skill checks for out of combat stuff than the full caster and bounded accuracy still helps him contribute at half or 1/4 effectiveness.
Or 1/16th or whatever...
Instead most of the out of combat stuff in the exploration pillar that full casters get to do he literally cannot do.
They hypothetical combat-incompetent caster couldn't attack 4 times in a round with a greatsword, either, but he can make his contribution of 'very little benefit' with his staff or dagger 1/round. So, sure, a single spell might solve a social or exploration challenge at one go, that doesn't mean the fighter had no contribution to make with his +1 survival or +4 intimidate or -1 diplomacy whatever, just that it wasn't called for in that instance.
That said I feel challenged now to actually try and see just how strong the non-combat fighter can be! I might be pleasantly surprised. I kind of doubt it though...
There's no reason a fighter couldn't be quite decent with a few skills by investing in the right feats. It wouldn't have the level of skill possible with those as with expertise nor the flexiblity and single-spell-solutions of a caster, but it'd be a more-than-very-little-benefit contribution when those specific skills or feats applied.