It damages verisimilitude that every fighter, of which there are presumably thousands in the world, all learn to be proficient in heavy armor and every weapon known to man, and then learn an ability to attack super fast once during a fight once they train a little more.
You can't seriously be arguing that strict classes enhance verisimilitude? That's, like, the #1 reason that people started point-buy and skill-based games.
Ok, now, your comment was "
There is no inherent value in having two characters with the same class be exactly the same.". That's what my confusion was about, certainly it make sense that two characters with the same class could be exactly the same- they could have been trained by the same master, for example.
Two identical characters is a bit different than every character! And yes, you're correct that if every Fighter was exactly the same, that would be a problem- but that's not the case and it hasn't been for a long time. As a previous poster noted, there are many ways you can customize your character, so that two members of the same class can be played very different.
For example, the Fighter.
You could build your Fighter to use ranged combat, large two handed weapons, two weapons, or weapon and shield. Your race, background, and ability scores could be different, as can be your fighting style. You can have different subclasses, and some subclasses, like the Battlemaster, give you choices of different abilities you can use. And playing a Battlemaster or an Eldritch Knight is a very different experience!
You could opt to multiclass. Or eventually acquire different Feats!
Your issue seems to be that it's strange for all Fighters to eventually gain Second Wind, Action Surge, Extra Attack, and Indomitable- a fairly small list of traits compared to all the previous points of potential difference. But I would counter that any Fighter would want Action Surge or Extra Attack, as they are incredibly useful abilities.
A purely ranged Fighter might not need Second Wind as badly as a melee Fighter, and well, nobody wants Indomitable (it's just terrible), so sure, there's a little room for alternate class abilities.
As for class-based vs. free form systems, there's advantages to either approach. If we're playing a free form system, it's possible for a player to make their character useless by investing in the wrong things, or to overspecialize to the point of ridiculousness. With a class based system, as long as someone puts their ability scores in reasonable places, I know that a 5th level Fighter gets 2 attacks and has an attack bonus of +5 to +8, no matter what else they did, which makes it way easier for me to plan encounters- if that damage verisimilitude, well, that's a sacrifice I can accept for the benefit of my sanity!