I think the problem with this argument and therefore the thread is that while the Fighter is unashamedly tooled for combat as a class, the nature of 5e means this doesn’t really matter.
Looking at the fighter class in isolation means yes there is a lack of specific class abilities to influence social interactions. Or to aid with exploration. However 5e doesn’t operate in isolation.
The fighter class is tooled for combat but a fighter character can be tooled for exploration or interactions. With 12 Charisma and the prodigy feat I can get +5 persuasion which is the same as that Charisma 16 Bard (except my proficiency will scale faster). I could do the same to gain proficiency with thieves tools and stealth, or survival etc etc. Backgrounds let you be great at whatever you like really and bounded accuracy means you don’t need to devote a whole character to an idea to be successful at it.
You may say that the fighter has to give up resources from background or feat to do this. I say who cares, you have to spend those resources somewhere. The trade off is I get to be ace in combat.
So because I can make a Fighter character who can interact and explore really well I don’t really care that the class doesn’t have those elements baked in.
At the cost of a permanent two stat points in a key stat or a significant amount of combat potential you can get a fighter character who has one single skill that will scale as well as a bard will
if the Bard doesn't really care about that skill (remember that Bards get expertise as a class feature). And if the bard doesn't choose to focus on persuasion they still cancel out your first two proficiency increases with spending their first two ability boosts on CHA, which they are going to do anyway. Meaning in the skill you have chosen to give up significant combat potential for you are either equal to or behind a not particularly interested bard until level 13.
But the Bard doesn't just get skills. They get spells that support what they do. Cantrips like Minor Image, Message, and Prestidigitation help. So do first level spells like Disguise Self, Charm Person, Comprehend Languages and Illusionary Servant. So do second level spells like Detect Thoughts, Suggestion, and Zone of Truth. And no, a Bard isn't going to have
all of those. But they can have some - and because combat is a simple process you don't need many combat spells. And on a day by day basis the Bard gets to decide where to spend their spell slots while the fighter's feat is permanently gone.
And I'm not sure what makes you think that your fighter can explore really well. They are in general good at climbing things - but not very good at spotting them, are frequently slow and not good at stealth, and don't have any real abilities to help. The Barbarian, for example, is significantly better in every way here.
All of which means your fighter isn't so much good at the social pillar as
not entirely incompetent. But well behind a bard who isn't even trying to be an expert at persuading.
As for "the tradeoff is I get to be ace in combat" feats are valuable. You gave up a significant amount of being ace in combat when you chose to permanently burn an entire feat to be able to play in another field at all. The tradeoff for the fighter getting to be vaguely competent at social skills rather than being the worst is that you cease to be actually ace in combat. This is a cost not paid by spellcasters.