Why are you trying to make what a PC says about themself into a game mechanic? You're right, we are not playing FATE. This idea that a PC says they are "the best swordsman in the land" has NO mechanical weight, and is not used to satisfy any game conditions.
I'm trying to make sense of what you're saying. And up until your most recent posts that was the best sense that I could make.
All it is, is what the PC has seen of themself within the narrative. That's all. If this PC has won every fight they have ever been in within the story... then they decide within the story to claim themselves to be the best swordsman in the land. And as a DM I find that perfectly acceptable. Does it matter to me if they are only 1st level?
But very little D&D play is going to stay at 1st level, and as the rules are presented an important aim of play on the player side is to go beyond 1st level. So the player knows that his/her PC is getting better, and generally in the fiction the PC is getting better.
Even in default 4e - that is, 4e that uses the Monster Manual as written and the tiers of play as described in the PHB and DMG - a character isn't plausibly the best swordsman in the land until upper heroic. (Because at that point anything better than them is a paragon or epic tier challenge and therefore not, in any ordinary sense, "of the land".) But 5e is more "earthbound" and less fantastic in those ways than 4e which I think makes this harder.
In 4e it's fairly easy to change the fiction away from those defaults - Neverwinter Campaign Setting does it one way (crunching paragon fiction into the mechanics of upper heroic tier) and Dark Sun a different way (extending paragon fiction into the mechanics of the epic tier). But 5e is less flexible in this respect also (and not accidentally so - it's a deliberate design decision to depart from the so-called "dissociated mechanics" of 4e).
Really? When does the game tell us the PC fights against a random town guard? I didn't realize that was a requirement of the rules.
For "random town guard" substitute "kobold soldier number X" or "orc warrior number Y" or whatever it is. D&D - in its published rules, and published scenarios, very much pushes towards multiple combats with multiple nameless foes.
The fiction of the story is about what actually happens... not what COULD happen.
<snip setting up of example of Arcana-skill rogue>
The Rogue PC out-Arcanas the wizards in this campaign story, and continually shows off his ability, even without having the best possible mechanics to do it. And why? Well, maybe the times when the Rogue PC shows off their knowledge of Arcana the player keeps rollings 17, 18, or 20s and the wizards NPCs roll 4s, 7s, and 8s. And the Rogue keeps winning these "contests" in arcana lore even without being the best mechanical representation. Even against wizards that might 2, 3, 5 levels "higher" than the PC. In the story those "levels" do not matter, the actual Arcana skill number doesn't matter...
...all that matters is that in-story, this Rogue has exemplified their superiority. And at that point... the DM does the proper improvisational "Yes, And..." and goes along with the idea that in this story in this campaign, this random Rogue PC apparently is the most knowledgeable magical-lore person in the land.
But unless the GM (or player?) is manipulating those dice rolls then this can't be planned for, and can't really be built around. Because it's luck.
this is all baseline improv rules. 'Yes, And...' and all that. The player says "I want my character's story to be that they are known as the best swordsman in the land." And I go, "Okay! Yes! Sounds great!" And at that point, I begin working stories that help exemplify that narrative
If the narrative depends in part on rolling repeated 16+ against others' repeated 5-, I don't see how that is going to work (assuming the game sticks to D&D's resolution mechanics).
the mechanics do not dictate the in-game reality of the story.
I really think that you are not explaining how your vision of D&D play fits with such basic features of the system as
characters gaining in level,
multiple PCs who have different strengths reflected in different mechanical features of their builds,
default lists of antagonists which consist primarily of mechanical specifications of said antagonists, etc.
Even in 4e your vision would be hard to pull off. I don't see how it works at all in 5e. Especially when spells are factored in, which is something I've raised several times now
but you've still not responded to.
EDIT: Apologies, I saw this:
Because the actual descriptor doesn't matter. It's whatever a character claims about themselves in the fiction. The whole "best swordsman in the world" thing was just a single example. By the same token, a wizard who is known as the greatest wizard known to man in a particular narrative I am running does not need to have better "game mechanics" than any other potential wizard in the game. Because up until those two characters actually start rolling dice at each other... those mechanics are meaningless. As a matter of fact, I don't even need to create game mechanics for that wizard and still let him be known in the narrative as the greatest wizard known to man. Because if it's known as such in the story... then it's known as such in the story. And I don't need to build this NPC mechanically to "prove" it. What a waste of time that would be. Build every single NPC mechanically just so we can categorize and compare who's the "good" blacksmith in the town or the "best" sage. Rather that just describe them as such.
If at some point down the road a PC wanted to get into a wizard's duel with that wizard? Then sure, I'll probably make the character up mechanically. But that doesn't mean I have to make him 20th level in order to define him as actually "the greatest" wizard either. I'll make him whatever mechanically would help benefit the drama of the story we are telling.
I'm also finding this hard to follow. Most of it seems to be about NPCs, which is fine as far as it goes (what you say about doing NPCs primarily through GM narrative fiat is fine for (say) 4e D&D or Apocalypse World, but not for (say) Runequest or Burning Wheel). But I thought we were talking about PCs.
If, in the fiction, everyone knows that great wizards can walk the planes, then how can a 1st level PC possibly present him-/herself as a great wizard? Your fighter example trades in part on the fact that combat is resolved by dice rolls (to hit, damage) and related mechanics (hp tallies) that don't correlate to anything particular in the fiction. But magic isn't resolved the same way (eg via Arcana checks).
Again the version of D&D to get closest to what you're advocating here is 4e but even it didn't go all this way.