But unless everyone at the table does do that the fighter will be the best. Also you are assuming the fighter is not using subclass abilities, feats or fighting styles to further his abilities in this regard.
"If no one else tries, you are the best" Okay, sure.... but that doesn't mean you are doing any good. Because all it takes is a single other person to try to throw you aside.
And, yeah, talking about the Fighter base class tends to focus on the fighter base class. I am not going to assume feats, because they are optional and the fighter might want other feats. Besides, most of the feats you asked for are giving spells, which kind of defeats the purpose. I am not assuming a specific subclass, because that only helps if you happen to play that subclass. Also, other than taking spells as an Eldritch Knight, you are only looking at Battlemaster manuevers (which those skill manuevers are hard to justify. They are only useful some of the time, and cut into the Battlemaster's combat options which are useful far more often.), Cavalier... gaining proficiency?, Banneret Expertise (Note: Banneret is considered one of the worst fighter subclasses because of it s other abilities), Samurai gaining profiency and then getting a boost with elegant Courtier at level 7.
So, I guess, the Samurai at level 7 is pretty good at Persuasion. That's it.
The Bard, Warlock, Sorcerer, Paladin... all of their subclasses are just as good as the Samurai, until level 7, and some of them (the Bard) wreck it, and exceed it even at level 7.
Wizards are the premier melee characters if (and only if) they design to that specifically.
A Wizard optimized for melee to include melee specific spells, contingency, upcast false life, etc will mop the floor in melee at most levels with an equivalent fighter optimized for melee.
However that Wizard will be worse at the other two pillars than a fighter who chose to invest in those and no better than the melee optimized fighter.
Really? What subclasses is that wizard using? Oh, it doesn't matter? Okay what feats are they using? Oh, it doesn't matter? Okay, well they did end up locking in, what, five spells? Let's say 10 spells devoted to melee. Contingency is at 11th level, so the wizard with no defined feats, no defined subclass, only has a mere 16 other spells they can take. Oh, and 4 cantrips
So how many exploration spells can they get with that? Let's say 8 exploration spells, things like Fly, mage hand, light, find familiar, that make them really good at exploration, and let's say they take some social spells too, things like Minor Illusion, Comprehend Language, Disguise Self, Borrowed Knowledge, Enhance ability.
And of course you still probably have some room for some AOE and ranged spells, right?
So, let's just take the basics. THe Wizard is better at melee than the fighter with those spells and build choices you left vague. They can also cast Borrowed Knowledge and Enhance Ability to get 1 hour of profiency and advantage on any check they want, make them a diviner wizard so they can swap a bad die roll out. They have a familiar and maybe even Arcane eye for remote scouting, the ability to bridge language gaps, probably invisibility, AOEs, access to light and remotely able to interact with the environment to avoid traps...
How is the fighter better at them than social and exploration stuff again? Proficiency and a high score?
It is a team game and the fighter class can be good at the social and exploration pillars while still being good at combat, because the fighter has built in mechanics that make it good at combat regardless of choices. Most casters don't. Outside of a Bard or Ranger a caster class who makes choices (specifically spell selection) to be good at both the social and exploration pillars will be pretty weak at the combat pillar. They can manage. Bard and Ranger are the only two that can pull it off.
No they aren't. Casters
brutalize combat. They need to build to handle melee combat, sure, but they have strong ranged options with a single cantrip, and then they simply need things like Hold Person, Sleep, Hypnotic Pattern, Hideous Laughter. Seriously, I had a DM who almost wanted to ban Hideous Laughter, it is a level 1 spell and if it lands the enemy is completely incapable of acting. Sure, they can save with advantage if they take damage, but you can just point and take a powerful foe off the board. Blindness can ruin an opponent. Have you seen someone set off a synaptic static dealing fireball damage while also debuffing the enemy with a -1d6 to all attacks, checks and concentration saves? Which by the way, stacks with BAne that is a -1d4 to all attacks, saves and checks.
I personally was told by a DM once that a fight would have led to a TPK for my entire party... except I was playing an artificer, and I cast web.
Every fullcaster is a menace in combat, and out of combat. By just picking a single spell.
