Aragorn, who is just a really good swordsman
I think that you are underselling Aragorn. As the host rides out from Minas Tirith to the gates of Mordor, Gandalf observes that "there are names among us that are worth more than a thousand mail-clad knights apiece." Even allowing for a degree of rhetorical excess, Gandalf is weighing Aragorn as the equivalent of 100s of ordinary soldiers. In D&D that should mean the ability to cut down whole swathes of ordinary combatants (some form of AoE or the classic one-attack-per-level vs men-at-arms), the ability to break the line (eg the fear effect of the original Oriental Adventures samurai and kensai), etc.
Even considered just as a warrior, a 20th level fighter seems to me to be really very limited.
I think it's a fine stopping point. You're at a Captain America-esque "peak of normal human condition" and can only get stronger and faster and tougher supernaturally or with equipment.
<snip>
If you want to play a warrior with supernatural ability, that's not the fighter class.
The game says one thing isn't magic and one thing is.
<snip>
If they don't want people to apply realistic expectations to something, then the game shouldn't claim those things are realistic.
You're playing the wrong game, pick up a superhero game. Level 20 martial characters in D&D do not have fantastical abilities simply by the fact that they're 20
As far as "playing the wrong game" and "not the fighter class": Moldvay Basic gives Hercules as an exmaple of a fighter, just as Merlin is given as an example of a magic-user.
As far as "the game claiming things are reaslistic" and "level 20 martial character do not have fantastical abilities": the game doesn't claim that certain things are reaslistic. In the 4e PHB, the abilities of martial PCs are expressly called out as going beyond what ordinary peope can accomplish.
Having to play a spell-caster in order to play a superheroic or preternaturally capable character is itself an issue, of flavour and of mechanical balance.
Not to mention: in the real, mundane, world, it is possible to kill or disable someone with a single sword-stroke. There is no reason, as far as realism is concerned, for spell casters to be the only D&D characters able to circumvent the hit point system in combat.
It seems to me that, in 5e, if I had to fight one fight, and had my choice of bringing along a full-strength 20th lvl fighter or a full-strength 20th lvl wizard, I'd be better of with the wizard. My preference, in a system with assymetric resource suites, is that for a wizard to match a fighter in combat the wizard
has to nova. Thus, a fighter is never a worse choice as someone to bring to a fight.
The balance for wizads then lies in their utility, non-combat abilities.