Something has gone wrong here, I think, because arguing from the specific to the general is how most knowledge in the social sciences is created (maybe economics is an exception, at least purporting to have a methodology closer to some natural sciences - that said, biology and some parts of chemistry might be closer to social sciences than to physics as the dominant conceptual model for natural science reasoning).
There's an element of pedantry in this post, but it's not just pedantry. "Story now", "step on up" etc are ideal types, intended to allow us to group phenomena in ways that are illuminating and explanatory (and that are not self-evident independently of deployment of the types).
The phenomena in question are RPG play experiences.
Can we have a RPG play experience where player protagonism drives the action (whatever exactly that consists in), where the principles that the GM follows are about following that action and piling on the pressure, where there is no "the story" because the new fiction the GM introduces is immediate and responsive in just those ways, and where everyone also knows that, when the fiction reaches a particular point in time, the protagonists will die? I don't see why not.
You might want some way to manage the passage of time, in order to make the coming of the end more than just GM fiat: some canonical way of mapping actions and events to the passage of time, perhaps a certain number of turns each, or even something like a Doom Pool. The timing could also be handled player-side: eg every PC has two Beliefs, and a tick is placed against a Belief if it is enlivened/engaged/implicated by an episode of play, and when every PC has a tick next to each Belief the next GM move is to bring things to their climax.
I have, but haven't yet read or played, My Life With Master, and I imagine it has some interesting and powerful techniques for handling this sort of thing.