This is an argument about taste. It's subjective. It comes down to verisimilitude: what feels believable to people in their stories.
Let's go with the example of the 100' leap that was proffered earlier as an example of what a fighter should eventually be able to do baseline. I'll add another: being able to lift a car, or weight equal to a car, over their heads.
For me, those are firmly in superhero territory. If I see that in a movie, I need some kind of explanation. I'm not just buying it as something people can do. If Conan suddenly leaps over a 100' gap, verisimilitude is out the window. Although it's a fantasy setting, that does not mean that anything goes.
Or take the first Guardians of the Galaxy movie. We have Peter Quill hanging out with a bunch of superheroes. When Drax can crumple a robot in his hands like it is tissue, I'm fine with it, because we've established that he's a super strong alien. If Quill did it, I would need a reason or I'm not buying in. When Peter jumps long distances, it's fine because we see his little jet boots go off, or whatever. I don't need the schematics for them, I just need to know that they are there, so that I'm not taken out of the story.
However, the film finishes with a bit of a mystery - it seems like Peter is able to handle an infinity stone just through the power of friendship. And thus the second film establishes that there is a reason for this: he's not as mundane as we thought.
Batman is generally presented as a regular human capable of super heroic feats through intense training and mechanical contraptions. Again, if we look hard at this, it seems very unlikely that an actual human could achieve some of his feats. But the writers and artists generally try to maintain that conceit. If Batman just randomly picked up a car and threw it at someone, we would assume that he had added a powered exoskeleton to the bat suit, or had otherwise acquired extra-human powers. If he jumps a 100' chasm, we see the batwings unfurl so he can glide. If he did it in his underwear, we would be puzzled, and verisimilitude would be lost - we would have fallen out of the story. In The Dark Knight Returns Frank Miller goes to great lengths to show how this mundane person could possibly fight Superman.
For me, and evidently for a lot of folks, it is important that D&D keep room for characters who are mundane people. Who can keep up, right to level 20, while not needing obviously supernatural powers. Quibbling about what feats or abilities, etc. logically have to be superhuman, and implying that therefore our taste is subjective and wrong, is missing the point entirely. Everyone's taste is subjective and wrong, much of the time, to everyone else. In a game targeted to mass appeal, like 5e, the priority is to make it work for the majority.
The 5e fighter patently does that. It is, by a wide margin, the most popular class in the game. It is widely considered a strong class, as demonstrated by its popularity and persistent placement towards the middle and upper placements in class tier rankings. Many of us have pointed out that we see it working well in our games, and in actual play shows. It scratches the itch it was designed to scratch, not for everyone, but for a whole lot of people.
And the thing I find most curious is that this discussion keeps ignoring the fact that there are plenty of options for those who want to play a fighter and have supernatural abilities. You can be a pure fighter and blink across a 100' chasm all day long as an Echo Knight. You can be a Psi Warrior and lift ridiculously huge things over your head and make giant leaps and such. It's already in the rules.
What this argument comes down to is folks claiming that their subjective taste is not well reflected in the rules, fair enough, but also then spending dozens and dozens of pages explaining, often very condescendingly, why their subjective taste is actually objectively correct.
