I don't disagree that class to some extent functions as class-as-game-piece, I think that's quite obviously true. But I would disagree that this is it's primary function because without the framework or example provided by the fictional (and historical) exemplars the game piece is a cypher. There is no recognizable game piece function without the exemplar, at least not one that's recognizable as a D&D class. I think I might use the word synergy to describe the relationship between the mechanical game-piece elements and the skills and evocative dressing that comes from the fictional inspirations. It seems to me that the end product was greater and more engaging than the sum of its parts.
I think I'm not as sure that the original game pieces needed a
particular trope or exemplar to work. Heroes and Superheroes can be Aragorn, Eomer, Conan, Lancelot, etc without worrying too much about details beyond that. And all we need to know about Wizards, exemplar wise, is that they use magic. The artillery-like effects are a product of wargaming requirements and conventions, plus maybe a dash of Dr Strange.
The role of classes, and their relationship to tropes, has changed a lot since Chainmail; and beginnings of that change can be seen even with the Druid and the Monk. I still think that, in the original conception of the game,
game function was primary.
I generally agree, for example that RPGs need resolution mechanics and a framework for deciding on the consequences of actions. The term framing is a bit sticky though. Not because it's not useful, but it often gets a bit kludged up in the narrative/not-narrative argument. Setting that aside, I think it's trivially obvious that what GMs do is frame scenes (or encounters, or moments, or whatever one chooses to call them) and allows the players to respond. This is the essence of RPG play, of the conversation that drives recursive alterations of the diegetic frame.
However, I also think that the above is perfectly possible without any reference to specifically narrative terms like rising action. Is that an aesthetic choice? Probably, yeah, I think that sounds correct. What that tells us though is that the options involved in that aesthetic choice aren't fundamental to RPG play, but rather interchangeable.
What I'm sensitive to - most recently, I was commenting on it in relation to Mythic Bastionland - is when a RPG
doesn't tell the GM how they should be making framing decisions.
Classic D&D does (although it doesn't use the language of framing) - the GM frames by reference to the location of the PCs on the map, with reference to the key; and also by reference to the outcomes of wandering monster rolls. Burning Wheel does too, albeit it's rules for framing are completely different from classic D&D. Apocalypse World is a little bit more indirect, but does set out a method for framing - the interplay between the prep of fronts, and the making of GM moves.
But the GM-as-world-mediator approach, at least when explained by its proponents, tends to eschew giving an account of how the GM is to make these framing decisions. It doesn't want them to be done by reference to rising action, or theme; but what approach
should be used is not normally explained.