A few final thoughts about this whole question, FWIW:
I’ve suggested that a Phantom Ideal Fighter underpins arguments for class-realism.
Aside from what I consider the most fundamental error - treating an abstraction as though it were a concrete thing - we know that "Fighter" is a category created by game designers for play purposes. The reification move takes this category and grants it independent existence - first in the player’s head as a design concept, then into the fiction itself. But the class “Fighter” is a sorting device, not a natural kind.
If you accept the Phantom Ideal Fighter as a real thing in the fiction, you immediately face the problem of why the boundaries fall where they do. Why is the Paladin a separate ideal from the Fighter? Why does the Ranger occupy a distinct ontological category? These divisions are transparently the result of design decisions - Gygax needed the classes to feel meaningfully different at the table, to give players distinct play experiences. There is no natural joint in fictional reality at which "Fighter" ends and "Ranger" begins. The phantom ideal smuggles in the assumption that these arbitrary design seams correspond to genuine divisions in the world.
The Phantom Ideal Fighter is invoked to bridge the gap between the mechanical class and the fictional warrior. But it doesn't actually bridge that gap, it just inserts a third thing that has the same gap on both sides. You still need to explain how the mechanical Fighter maps onto the Ideal, and how the Ideal maps onto the fiction. You've doubled the explanatory burden without discharging any of it. This is structurally identical to the homunculus fallacy in philosophy of mind: explaining vision by positing a little man inside the head who watches the images, which just pushes the problem back one step and generates an infinite regress.
To take the Phantom Ideal Fighter seriously as an ideal is to commit to a fairly heavy piece of metaphysics: something like Platonic forms. There exists, in some sense, the Form of the Fighter, of which actual fighters in the fiction are imperfect instances, and of which the mechanical class is a representation. This is an enormous philosophical commitment to smuggle in through the back door of a game rule. And, really, it's Platonism done badly: real Platonic forms are arrived at by reasoning about essential natures, not by reading a class description that was written to differentiate play styles.
The Phantom Ideal implies that there is some essential Fighter-nature that all fighters share: some cluster of properties that makes something really a Fighter as opposed to merely fighter-adjacent. But try to specify what that essence is without just redescribing the mechanics, and it immediately evaporates. Is it martial training? Barbarians have that. Weapon proficiency? So do Paladins and Rangers. Lack of magic? Not in most editions. The supposed essential nature of the Fighter turns out to be either empty or just a redescription of the mechanical chassis, which proves the point: the "ideal" has no independent content. It's a shadow cast by the rules, not a light source.
There's a peculiar asymmetry at work. People who claim class is real in the fiction might say "my Paladin wouldn't do that," using the class as a constraint on fictional behaviour. But they rarely push it the other way: they don't argue that because their fictional warrior is brave, honourable, and skilled, the Paladin class must have a Courage stat. The Phantom Ideal is used selectively: to import mechanical constraints into the fiction, while the fiction's own logic is never allowed to revise the mechanics in return. This asymmetry reveals that the mechanics are doing all the real work, and the "ideal" is just the name given to the direction of flow.
Perhaps most damningly: the Phantom Ideal Fighter cannot be described in purely fictional terms. Ask someone who believes in it to tell you what the Ideal Fighter is as a person in the world: their training, their social role, their psychology, their history. The answer must either be something so generic as to be trivial (a person who fights well), or it translates back into mechanics (well, they'd have high Strength and Constitution, they'd be proficient with most weapons...). The moment you try to look directly at the Phantom, it either disappears into banality or reveals itself to be the mechanics wearing a fictional disguise.
I don’t think there’s a “right” way to play D&D, but I’d consider myself quite old-school in many of my sensibilities. As far as I know, Gygax never seems to have explicitly addressed the diegetic status of class - whether it "exists" in the fiction - as a philosophical question. That particular framing is really a later RPG-theory concern and Gygax predated that conversation (and largely showed no interest in it). What he did articulate - and quite forcefully - was something adjacent. In a long piece in Dragon magazine issue 16 (1978), he argued that D&D was a game first and foremost, not a simulation, and that the mania for "realism" that dominated the hobby in the late 1970s was a category error, worshipping what he called the "false deity" of realism. His core claim was that "as a game must first and foremost be fun, it needs no claim to 'realism' to justify its existence."
Gygax was defending mechanics against the charge that they were insufficiently realistic, which implicitly treats mechanics as modelling tools, not as descriptions of in-world reality. In the Gygaxian view, class, hit points, AC, and so on are all just game apparatus; tools for making play fun. He was hostile to the conflation of game mechanics with simulation of reality, and he seemed to think of class primarily as a player-facing tool rather than a feature of the imagined world.
The irony is that the "class is real" position is in some ways a byproduct of players taking D&D more seriously as a fictional world than Gygax himself apparently intended.