Why do they need to be the best at these things to allow them to contribute?
There's contribute in the balanced team-play sense, and there's 'shine' in the spotlight-balanced sense. Out of combat, any warm body can contribute with the occasional untrained/modest-stat d20 check; in combat, any body (it needn't even be warm, it can be a necromancers animated minion, for instance) can contribute by taking up a space where an enemy might optimally have wanted to stand, soaking up some attacks, and/or swinging and hoping for a high roll - that's the wonder of BA.
In combat, every class can offer more than just a temperature-optional-body contribution without even trying - every class gets decent hps & proficiency in a fair selection of weapons, and/or adequate enough combat cantrips as an at-will combat baseline, and plenty more on top of that depending on the class and the choices they made at chargen/level-up (or even that morning for the prepped casters). Every single class.
The same is very nearly true of the other two pillars: every class has some perks in either exploration or social (if not both), most have some flexible resources that can go either way (including prepped casters making that choice after every long rest), and if they do lack much of anything in one remaining pillar, a background might cover it to an adequate degree to get by - you're only sitting out one a pillar of you intentionally neglect it.
Except for the fighter & barbarian, they get virtually nothing from their classes outside of combat ability - they can use their Background (as can everyone else) to get a little modest ability in one other pillar (you can take Outlander to be better at exploration in the wilderness, or Criminal to be better at skulking around and opening locks in cities & dungeons, for instance), or compromise their builds to devote a better-than-usual score to a tertiary stat like CHA, but outside of that they're looking at warm-body 'contribution' some substantial fraction of the time (depending on the emphasis of the campaign, of course).
No class is ever that put out in the combat pillar.
The Fighter is permitted to increase his/her CHA. One may select feat to allow one to pursue/assist in the social pillar.
Yep, the fighter can put one of his bonus ASI's into a feat like Actor or a +2 CHA, or, hey, sink both of them into getting both. Again, that's the original point. Now he's maybe up to par in the social pillar, but he's not optimizing his DPR as fiercely, and he's got nothing left to invest in exploration.
In general, I agree that Charisma is an unusual choice for a Fighter, and it probably shouldn't be an ability score given priority since it does a lot less for a Fighter than it does, say, a Sorcerer. SOME Fighter builds can benefit from Charisma, though to a lesser degree than other classes.
It really would have made a lot of sense for the fighter to get some non-combat features that keyed off charisma, because, in concept, the fighter is easily the most-relateable class, as has been eloquently argued, already:
Except, of course, that in-play many of those other classes are going to find themselves strongly prejudiced against in many (but not all) social situations.
The primitive barbarian savage, demon-worshipping warlock, thieving rogue, meddling wizard, bestial druid, self-righteous paladin, unworldly monk etc., are all facing an uphill battle no matter how great their CHA is.
It would hardly be out of line for fighters to attract a band of followers like they did back in the day - and, under BA, they'd actually be /useful/ for the first time.
