You can't really say that emergent gameplay is the same as "story." Stories have characters, protagonists, antagonists, pacing, arcs, dialogue, etc... Stories can be told and re-told. In theory, it IS possible to create a story through emergent gameplay. However, most modern games do not then save the story to be retold. It is a story that is particular to a single experience had by a single person. (or in the case of multiplayer games, a group of people) These kinds of stories are also extremely simple. They normally involve very rudimentary interactions involving the mechanics and rules of the world, and the ways in which the player interacts with them.
This is veering away from story in particular, but we work with the building blocks of fiction regardless of how much story is "designed" into the game structure. Characters can be developed through game mechanics alone(example: the Pac-Man ghosts) and the experience of these characters can be shared, even if the specific story is somewhat different. The tendency of games to combine "plausibly exaggerated" experiences(wacky physics, overpowered abilities, etc.) with simplistic motives(beat the bad guys, save the princess) could therefore be construed as a means to help along a player-internal narrative, even if it holds no water as a non-interactive story.
Imposing story-by-design constitutes a shift from the player-internal narrative towards a takeover of all storytelling functions. The main issue presented once you do this is that the internal narrative never goes away in an interactive experience, so you actually have two narratives running in parallel, each one stating "This is why these things are happening." If they clash, they both lose power.
But in such cases, the mechanics are more readily believable, since(as long as a bug doesn't make them break) they are shown to be self-consistent all throughout gameplay, while non-interactive story elements are presented in arbitrary fashion, with no room for player experimentation.
There's an interesting distinction here - a character with an extremely detailed backstory has no contradiction, as long as the backstory resurfaces through mechanics like attributes and powers. But a story layered on top of the mechanics, where scripted events override the rules, is going to be more fragile.