So, this is where I answer my own question. I'm echoing a few sentiments that came mup in my comments, but also differing from some of them
The question is what stories RPGs tell, or more specifically, what story elements RPGs excel at. I'm not saying that these are limits on RPGs - it is a nearly unbounded form - but as with every medium, there are things that it excels at and things that it's less strong at. With that in mind, I'm not looking to provide an encyclopedic list of things RPGs can do well. Rather, I'm focusing on the things that an RPG would be the first choice for.
The first and most obvious element is agency - players in an RPG may make choice to impact play. This is almost unique to RPGs - there is some overlap with elements of improv (a reason many RPGers look to improv for inspiration) but it is a different sort of animal if only in terms of the framework it exists within. This is kind of awesome, because agency leads to investment, but in and of itself it a hollow thing, like saying painting lets you use colors. It's true, but it's kind of dwelling on a unique tool, not a unique result.
There may be some fertile ground around the idea of investment, but I admit I haven't really found it yet. Like agency, it seems to be a tool, but there are some unique manifestations of it. It can take the form of shared knowledge of a truly staggering scope but I'm not sure how much of that is unique to RPGs. The shared knowledge of the Forgotten Realms is staggering, but the yardstick to compare it to is Middle Earth, something born of novels.
Creation? Perhaps one of the strengths of RPGs is the profound blurring of the line between author and actor. But if so, what can we do with that?
Maybe it's something more obvious: what about Play? After all, if RPGs are art, they're art you can win. That feels closer, if only because it's definitely a unique reason you'd want to play an RPG rather than write or create in another medium. Heck, compare it to writing contests: novels may make for better stories, but RPGs make for better competitions! But again, I'm not sure that suggests anything unique about what they say.
The thing I keep bumping my head against is that I can think of a number of fairly unique techniques to RPGs, but those are useful only after I have decided to use RPGs as my medium. They don't suggest reasons why I'd choose to tell a particular story with an RPG rather than in some other way.
But there's one exception. The element that I think might actually be the most important is reflection - the events of play reflect back on the player. In other media, you may become invested in the characters, but in RPGs, the characters may become invested in you (or at least your character). The creation of a reality that looks back on the actor is huge, and it's often dismissed as mere sleight of hand since these people and places are not real, but I would counter that the fact that they are fiction does not rob this of its power. Most creative media rests on the idea that we may be powerfully and truly moved by fiction, and I see no reason to carve out an exemption here.
And that suggests that the reason I'd want to use an RPG, rather than a book or a movie or a play or a painting is if I want it to be your story.
That's some powerful mojo, but it has a few implications. First, it underscores the fringey-ness of the hobby. A game doesn't say much to the people who are not playing it (as any number of recorded sessions will tell you) but that's because it's not supposed to. The more it speaks to the world, the less it speaks to the table. That's awesome for play, but it also means we're unlikely to end up with our games hung up in some equivalent of the louvre.
Second, and perhaps of more immediate consequence, it demolishes most any sense of ownership or authority in the creation of play. A GM may do everything he can to make play awesome, but if he creates his story, he's misusing the medium. It is only by surrendering that power to the players that he will really succeed. This can be a really, really hard thing to grasp, and it can come a s slap to the face to a GM who busts his hump making play rock, but there it is.
There are some further implications of this, but they enter into the realm of conventional wisdom. The insights that our hobby isn't scalable or that the GM shouldn't treat players as an audience are far from new. But I wonder what happens if we embrace them as weaknesses of the medium and try to focus on the strengths. I imagine it looks weird, sure, but at the same time I sometimes suspect that we're trying to write novels with paintbrushes, and that we'd be a lot better served deciding what we're actually doing, and pursuing that with passion.
1 - The great pain is that "Telling a Story" does not make this list (or at least telling a specific story). Books, movies and plays all do a better job of this for reasons I hope are self evident. Yes, you can tell a story with an RPG, but you need to jump through more hoops than you would if you were just writing it.
2 - Though speaking of which, you know what I'd buy a book on to see how it applied to RPGs? Editing reality TV. Those guys are really good at creating a narrative out of a bunch of stuff that happens and putting it all together like that's what's actually going on. It's masterful fiction, and it resonates a lot with the idea of stories being created as a result (not an intention) of games.
3- We've all seen the GM who should just be writing their book.
4- Though I amuse myself at least in the route that lead to those points this time.
5 - I feel like this is also dovetailing into my growing certainty that setting is king, but I do not trust that conclusion quite yet.