What exactly is the gap trying to be filled here?
A story is characters doing stuff and stuff happening to characters. Are characters not doing stuff or having stuff happen to them?
Well, I mean, if all we define story as is "whatever happens to the PCs" then what's the point of any discussion of what happens? Obviously the happening must have certain qualities which can be attributed to it, right? I think it is more a question of whether the 'story' in your sense is or is not in any sense dramatic or interesting. Is it simply a long rambling list of scenes with no unifying theme? Do the characters have some sort of substance to them, or are they simply 'pogs' in a game focused on some other goal like 'who can get the most GP' or something like that?
So, what Dungeon World does, as an example, is to set the scene, the world is fantastic, its dangerous, the PCs are unique individuals, and some sort of 'stuff' is happening in this world (which is independent of, but will inevitably have an effect on, the PCs). Also whatever this stuff is, it is chosen to correspond in some degree to the avowed interests/character traits of the players and their PCs. The PCs have traits, like bonds and an alignment (kind of a primary personality concept) that the players will engage with, because they get XP for that. Whenever the PCs roll low, whenever the players don't know what they want to do next, or whenever the GM is given a 'golden opportunity' (IE if the PCs ignore some obvious danger) then the GM makes a move, usually a hard move. The GM may also make soft moves, which just draw doom closer (IE your 2nd to last torch just guttered out, and you're a mile underground). The upshot is that PCs are always getting into something, and its always got an element of trouble to it!
So, its not like there IS a story, certainly not explicitly, but it is true that DW, unlike D&D kind of cannot function when something interesting isn't happening, and if that situation arises, the GM literally hits the party with something, immediately. There isn't ever the possibility you are just putzing around or that the PCs have to go to the action. I mean, you could decide nothing happens for a while, but that sort of period would be elided in DW. You wouldn't normally talk about how the halfling makes dinner, unless it has some potential to 'go somewhere'.