How about this. Run a simple test. Grab 20 people walking in a downtown area, and show them a picture of Conan, King Arthur, and Boromir. They don't even have to be labelled. Ask them which one is a barbarian? I bet you 20/20 get it right.
Depends on which pictures one uses, surely. A Conan wearing the crown of Aquilonia...
VS Viggo Mortensen in slightly-grubby full Strider outfit...
Would surely make it a significantly tougher call than 20/20. King Arthur is of course a harder call, since there isn't a single actor or performance to whom one can point and get a "yes, THAT man is Arthur!" like you can with Viggo or Arnold. But, surveying the first few images from a simple Google image search, using "King Arthur actor" (no quotes in search) to avoid all the much older
paintings and illuminated manuscripts of him, this one pops up from an apparently forgettable 2004 film...
Seems to me like those would be a lot harder to distinguish from one another. They
all look like Fighters to me. And for a younger audience that may not recognize Conan the Barbarian if he isn't shirtless and carrying his Atlantean sword (remember, the Arnold film came out in 1982; anyone under 45 today could not realistically have seen it in theaters), it seems extremely overconfident to assume that 100% of people in a random sample from the street would instantly know he's a Barbarian and not a Fighter or even a Paladin. Heck, I wasn't even
trying to do this, but I accidentally ended up giving Conan the only plate armor in the group!
My point being: your test is not as reliable as you think unless you heavily bias it with primed images. Those images may evoke certain archetypes, sure, and those archetypes are relevant. But they are not absolutely determinate. Further, as I'm sure some folks will note, it is somewhat humorous that you go all in for "Aragon is Fighter, Arthur is Paladin" when from a purely literary perspective it should actually be the reverse: Aragon actually DOES have some magical talents, albeit usually mediated through herbs or magical substances, while Arthur is never shown to have any magical abilities in his lore (given "is the son of the king" is not magical itself, just recognized by magical things like the sword in the stone.)
Even if they have no idea who Conan is. Now ask them which one is a paladin. Provided they know the word's definition, I bet you 20/20 get it right, even if they do not know who King Arthur is. Now ask them which one is a fighter, and I bet 20/20 get it right. That is because words have connotation, and connotation is attached to tropes, and tropes are attached to lore. You could do the same thing with wizards and witches.
You are correct, Gandalf is different than Potter in a few instances. Most of the time they are the same.
Where's this "few" thing coming from?
Gandalf is still inherently magical even without his staff, it's just a useful tool. Harry Potter literally cannot use magic without a wand (learning to do so is an extremely advanced skill that almost no one can master). Essentially all Potterverse magic is spoken with specific incantations (again, it is a highly advanced skill to cast wordless magic in HP, neither skill is learned by any main characters, with only Dumbledore demonstrating either in any serious capacity), while Gandalf is almost entirely wordless and freeform. Magic in HP follows ironclad rules about what it can and cannot do (e.g. it can duplicate food that already exists or teleport food from somewhere else but it cannot make food just
appear from nowhere), while Gandalf's powers are pretty much whatever he would like to achieve, limited only by his Istari oath to act only as an advisor and helper (as broken by Saruman, or even Sauron, who is the same class of being as Gandalf). Magic in Harry Potter is a purely bloodline effect, with the muggle-born wizards and witches being canonically born from squibs (essentially "wizard-born muggles," people who carry the "wizarding" gene but can't actually use magic) marrying into muggle families due to prejudice against them in the Wizarding world, while magic in LOTR is basically the exclusive province of incredibly powerful immortal beings (maiar, elves that actually visited Valinor) or the setting equivalent of liches (Ringwraiths). Even visual comparisons are weak; Radagst the Brown is clearly a Druid in D&D terms, and there's a pretty dramatic difference between Gandalf the Grey and Saruman the White, while HP characters all evoke a slightly cleaned up version of the black cloak/peaked black hat of stereotypical European witches. There is no even moderate possibility of a child using magic in LOTR, while in HP it is not only commonplace, demonstrations of instinctive magic as extremely young children are the hallmark of actually being a witch, and are looked for most earnestly by parents who fear their children might be squibs.
Are you talking about characterization? If so, then I agree, you can have a spectrum. But that doesn't negate a class's connotated trope, which contains lore.
Characterization is certainly part of it, yes. I'm referring to the whole package: visuals, behaviors, themes, context, in-setting mechanics, all of it. The best you can argue is that there's a bundle of stuff associated with each term, and that various different properties or works instantiate some particular subset of that bundle. There aren't enough universal elements that perfectly signify what's a "barbarian." There are a lot of components that can work, sure. But many folks would identify William Wallace from
Braveheart as a Barbarian (woad, bit grimy, rugged, light armor) when he's much more a Fighter or even a Warlord in his actions and context. Gandalf (the White, anyway) is much more like a 5e Light Cleric than a 5e Wizard, and the other Istari "Wizards" we meet similarly fail to conform fully to the whole list of D&D wizard tropes. If there were a wand based Warlock patron, I would actually say HP witches and wizards are more like
that than they are like any version of D&D wizards (except
maaaaaybe 4e), what with so many of them having what are clearly familiars (Hedwig, Scanners even if he turned out to be a person in disguise, Neville's toad, Crookshanks, etc.) and Hermione being so obviously a Tome Warlock.
Yes, there are bundles of ideas, and yes, these ideas matter. But even in your example cases, there's a huge amount of ambiguity and difficulty. Conan is a Barbarian in fiction, but he behaves more like a Fighter (and certainly in old school D&D would have been a dead ringer for Fighter as someone who was crowned by his own hand...even though many aspects of the Barbarian class were inspired by him!) King Arthur to you is clearly a Paladin, but to me he's more clearly a Fighter, since he never really practices magic in any of the stories I grew up with, that was always Merlin's job, or maybe a job for an actual priest. And Aragorn surely is a clearly iconic character who helped inform the bundle of characteristics associated with a class...but that class is Ranger, not Fighter, and yet of the three you've cited he is the one that most clearly fits Paladin in my eyes.
THAT is the problem I'm pointing to. People can easily look at exactly the same character and come to different conclusions about which class they are or should be. And even within a single class, people may not agree on which characters actually count as representative examples, even if in their own settings they're explicitly called by the same name.
Which leaves us where I said: yes, there are tropes and archetypes etc., and yes, those things DO matter, but they are rarely if ever so clean cut and ironclad that they exclusively determine one and only one way things should be represented. Instead, these things are polysemous, they are vast and contain multitudes, their borders are porous and their link to the use in fiction is often muddled or inconclusive. That does not, in ANY way, mean that these tropes and archetypes don't matter. They do. Sometimes a lot! But they also cannot be treated as inviolate barriers, because people don't
use them that way. Not in fiction, and not in TTRPG play.
Instead, they are treated much more like the rules of fashion or music composition or writing: useful guidelines, starting points, patterns that stand the test of time, but which cannot be relied upon to guarantee the best results always without question or thought. There is no formula for great writing, there is no algorithm for mellifluous music. That same structure is how class tropes and archetypes work; they are very functional and identifiable baselines that can, and
should, be defied
if doing so produces better roleplay. Which makes things messy and unclear. It makes judgment calls like "is King Arthur a Paladin or a Fighter? Is Conan still a Barbarian if he never rages? Are HP characters really 'Wizards' if their magic comes from their blood inheritance, with no amount of training able to install it in an ordinary human?" difficult and personal, something that cannot have objective answers because the answers are about what you believe or find the most utility in.
And it would seem that WotC is with me on this one. Consider the Barbarian and its subclasses. The Totem Warrior, Zealot, and Beast are all radically different takes on what it means to be a Barbarian, and they only loosely touch on what Conan does or looks like. The various "blend fighting with magic" options (like Valor/Swords Bards, Eldritch or Rune Knights, Bladesinger/War Wizards, arguably all Paladins, Bladelocks and especially Hexblades) show that there's not a sharp separation between those archetypes, even though most people intuitively feel the separation between them. (Indeed, I would argue that the very fact that people DO feel such a clean separation is why there are so many hybrid options, while there's far less of a felt separation between Fighter and Rogue, and hybrids between those are much less notable or prolific.)
Classes have implied tropes and themes. People actively modify, ignore, or replace those themes pretty much constantly. Even when people aren't doing that, there are often disagreements about which tropes or themes belong to a given class, and even more disagreements about which characters in fiction correspond to specific classes via those tropes and themes.