My answer to that is you can. Conan was blessed with plot armour. I don't know too many game systems where tossing a stool is going to kill anyone in one hit, let alone before a readied action can go off. As for Boromir he demonstrated tremendous strength, endurance and courage. When it came to it he single-handedly killed dozens of orcs before he was finally felled by massed archers, buying his companions time to escape. That sounds exactly like what would happen to a high level champion without magical backup in that same scenario. Where does the problem lie?
Really? You should look at a few non-d20 systems. Oneshotting someone with a stool is super easy in Fate Core, for example, with the math still being balanced. As for Boromir defeating dozens of orcs...that was possible in 3e and 4e, but with bounded accuracy, even a dozen orcs can be incredibly deadly for a fighter. A high level champion would go out like a chump in such a situation. A high level wizard would cast meteorswarm or forcecage or teleport or...well, you get the point.
Define powerful. A high level fighter is worth dozens of lesser men in combat. He can chop a warhorse in half in a single blow and jump off a building without even slowing down. He is about on par with Captain America and maybe even Wolverine.
Powerful. The kind of fighter that can, as you say, be worth dozens of lesser men in combat. Two dozen guys with simple weapons could easily overwhelm a fighter. The kind who can chop a warhorse in a single blow. A highlevel fighter does 2d6+5 damage on a hit, averaged at 12. A warhorse has 19 hp. The kind of fighter who can jump off a building without even slowing down. Any fall of 10 feet or greater inflicts damage and knocks you prone. In every single one of your examples, you name something that a wizard can do (fireball, sorching ray, featherfall) but a fighter can't. That's...that's kind of impressive, in a weird way.
As for being on par with Captain America...which Cap are we talking about? The 616 Captain America who can outfight Hulk, Spider-man, and just about everyone else? The Ultimate Captain America who uses 60-ton tanks as bludgeoning weapons? The cinematic Captain America who can take down a science fiction warplane in a single round? Yeah, I wouldn't mind playing any of those guys.
The problem is not that the fighter isn't powerful. It's that punching someone in the face is not as deadly as shooting them wih a LAW. So you're looking at Captain America and wondering why he doesn't get to use the cool toys. He does, he just has to get trained in them first, just like you can't fly a plane without training.
Flying a plane /= casting a spell
Seriously. This forced analogy that magic is equal to technology is ridiculous. It's an analogy whose premise is challenged immediately, and if all parties can't agree on that, it really doesn't have any weight in an argument.
Or he has to go out and get the cool toy. The DMG is probably going to include classic D&D items like flying carpets and brooms. Why would you give those to the Mage who can already fly? Kit like that goes to the fighter. And a fighter with a flying carpet, a +1 bow, and a barrel of arrows can kill the Terrasque without risk.
Whither balance?
Yes, yes, if you give the fighter magic items, he can almost do things like a wizard. This is silly.
First off, that's not how I interpret hp to work, but let's not go down that rabbit hole.
The point is, many people do want to play that way. Telling them it's unrealistic or silly won't get you far. So they get, by my count, 5 out of the 40+ subclasses in the PHB to work with. If you want a mythic supernatural warrior, totem barbarian, monk and ranger are exactly that.
There is a fundamental difference between calling something supernatural and mythical and it actually being supernatural and mythical. Yes, monks have mystical flavor, but they're weaker in combat than a fighter and their "supernatural" abilities are either inferior spellcasting of the least useful abilities or minor, super specific benefits like "you don't age, but you still die of old age" while a wizard has, again, hundreds of spells and can even become immortal through reincarnation.