I think I've figured out an example to show what I'm trying to say about gameplay not being a story:
Take a game. Strip out all the story elements, all the aesthetic tidbits. Everything with flavor.
Let's take, for example, Tetris. Tetris is pretty close to being without a story. The only thing we'd have to take out is the sound effects and the music. So let's do that. Imagine playing a muted "Tetris".
Now, here's the hard bit. Instead of "blocks", you're simply moving and dropping "shapes". These shapes have a distinct shape, but no distinct visual appearance. I don't mean they're flat white, or just an outline: those are visual appearances. I mean, they're just shapes.
What you are left with is gameplay.
You're stacking shapes in a way which will make them vanish.
Now, as you're playing it, you might be thinking, "ack! Getting too close to the top!" or "Damn, I didn't want to stick that there."
Those are the only "stories" gameplay contains. And those aren't stories. They're a linguistic spasm in response to a perceived reward or penalty. The only reason they exist is to hone our skill at dropping shapes into the right spot. They do not exist as stories, they do not have any aesthetic value. They are simply blurbs which vanish as quickly as they arrive.
You can build a story out of these. It's easy: "Man, I was so close to losing, but I pulled it off in the end."
But that is idealization after the fact. While you're playing, all you're doing is pattern management. Idealizing something afterwards is wholly besides the point. It has nothing to do with the gameplay, any more than telling a story about how you hit a tree with a stick has anything to do with the act of hitting a tree with a stick.
Now, if you're insisting that the little blurbs of "oh, that's not right" or "a little to the right" are "stories", then I'm taking umbrage at your desecration of the English language. If those are stories, then literally everything is a story, including this very essay. This very sentence, standing alone.
You're probably right: those little blurbs have something to do with stories. But I think it comes from the other direction. I think stories are an attempt to get people to feel those blurbs, rather than visa-versa. And those blurbs - those are just artefacts. They're just signs of the underlying pattern recognition process. I don't think "a little to the left", I simply know that I want it a little further to the left.
If you want to say that the underlying pattern recognition process is somehow based around "stories", please don't use the word "story"! Stories are accounts of happenings. "A little to the left" isn't an account or a happening! It's a blurb that tells you you've screwed up a bit. It's a memo on the sticky paper of your brain. It's gone as fast as it comes, and it's simply... not... a... story.
Please, am I making my opinion clear? I didn't realize it was going to be so hard to explain.
Addendum: Stories are fantastically awesome. They touch the player. They drive a game. They drive gameplay. But that doesn't make gameplay into stories.