I think focusing on "narrative" might be a bit of a detour that is a little too zoomed out to capture the thrust of things here. What we're really concerned about is the following superstructure:
* A goal with interposing situation/obstacle which must be overcome.
* The above relationship creates a deferred state of satisfaction while engendering discovery contingent upon the resolution of the above.
* From these dynamics a natural scheme of rising action > climax > conclusion arises.
* Whether its climbing a route, playing (or witnessing) a football game, participating (or witnessing) a boxing match, running/playing a dungeon/hexcrawl, running/playing a series of islands in Agon...string enough of these sequences together and an arrangement of story tends to emerge.
Now, the variables of nature of the goal, nature of situations/obstacles, what satisfaction we're deferring to post-resolution, what discovery we're deriving upon resolution, and the integration of all of these things via a novel set of procedures and scheme will create a unique experience. Whether that experience entails a compelling "narrative" or not is likely autobiographical.
But the superstructure for games yielding story (or at least our brains cataloguing it and experiencing it as such; there is evolutionary theory here and some evidence...but I'll leave that be) tends to hold.
Now...
Here is something controversial that I've said elsewhere. I think what constitutes "a game" vs "performance art" is interesting (and intersects with this thread). Much like choregraphed combat (performance art) doesn't defer satisfaction and entail discovery around the questions of "who exhibits the most prowess" or "do you have the courage to stand up and fight for x (your beliefs, your beloved, yourself, etc)", obstacles/situation and their resolution as performance art rather than crucible (illusionism approaches to play) also doesn't defer satisfaction and entail discovery around questions of prowess and ethos.
While it doesn't bear exactly on this thread, I think the question of "game or performance art" is an interesting one. Is the crucible component the pivot point of a game and that is why the arc/pyramid above inevitably emerges? And while performance art not done poorly definitely lends itself toward being a "story machine," the low stakes, choregraphed nature of it robs the crucible component so great care must be rendered in the choreography lest the deferred satisfaction and legitimacy of discovery will fail to launch. Going back to games now, I think you can see the same paradigm emerge when there is virtually no agency in the play due to the dynamics fo the game. For instance, if a game is "solved/resolved" after we simply answer the question "who goes first," then I think that game's claim to crucible, to deferred satisfaction, to legitimate discovery also fails to launch.