Not all RPs are "games" (in the sense of DnD or Skyrim). I would consider both Presidents and Andes RPs. People take over a persona, i.e. they play a role (of a person). So yes, RPs can exist in the AAR boards if they are using a Paradox game as a base.
Obviously a little bit of role-playing is inevitable in Interactive AARs. Even if it's just readers barracking for one party over the other, that usually involves a little bit of role-play. (even if that role is of a nameless party member or plebeian) I am not totally against role playing in Interactive AARs; I am against RPGs masquerading as AARs. When you have mechanics, like say coups in SoTA, you have a game, not a story. When the story itself is dictated by the players themselves, rather than the choices they make, you have a game, not a story. Most importantly, when you refer to your readers as players, you are hosting a RPG, not writing an AAR.
I would consider those aforementioned to be RPGs. I don't believe either would work very well as AARs in their current form if the "Interactive" components were removed. In contrast, a "Choose Your Own Adventure" story would still work as a story if the Interactive component was removed. It wouldn't be as fun, yes, but it would still work. This is the difference between an Interactive AAR (like "LTRCT") and a RPG (like the aforementioned).
I am not entirely sure whether a Character-based Interactive AAR would actually work without degenerating into a RPG. I suppose your Idea Mr. C. is a workable one, as the readers are prompting the author (or prompting one reader who prompts the author), rather than directly dictating the story to the author. Provided you can keep full control over the story, I think it could work quite well indeed. I do agree with TH though that AHD has a lot of potential for an Interactive AAR (in my view more so that EU), and I would like to see one for that particular game rather than EUIII.
That's a tough stance to take, since they're stories about games
But they are not games themselves. A book about a film is not a film in itself, it is a book. It's not a difficult concept.
So what would the players do under an absolute monarchy? The problem I foresee is that there's basically no player interaction at all early in the game.
Point and case.