I agree with a lot of what has been said, but I would also like to raise the example of Magikarp Jump. Its gameplay is so minimalistic that it openly tells you when things are random - literally the most significant features (fishing a Magikarp, point gains from training, the results of random events - these factors decide JP, and everything else in the game depends directly on them) are all randomized and require next to no user input. In fact, I'm pretty sure training still happens if you don't tap the screen? Tapping is just faster.
I don't really think people are playing Magikarp Jump for the gameplay - literally to tap on things that are hardly even time-specific (you don't have a limited time to tap on food, you don't have to tap quickly for the events, and once your partners have become usable, they don't disappear/go back to needing to wait until you've used them). Tournaments don't depend on user input, only on JP (which has already been established to depend on randomness), and if you lose, you can try again after some time - and generally, that "some time" is already what you needed to train Magikarp more (you're simply not going to do any better if you only try again right away), so it's not even a consequence, just a reminder to train. There are very minimal stakes, and user input is very minimal. Gameplay in Magikarp Jump is the simplest it could possibly be.
But then look at /r/Pokemon. It... it's very difficult to say that everyone has stepped away from this game and isn't enjoying it. I'm sure a fair few people got bored of the gameplay, but more than enough people have stuck around, and I started playing it because it was good enough that two friends of mine recommended it to me, separately, because they enjoyed it - and neither of them said a word about gameplay and functionality.
Now take away the cute dialogue, cute premise and general silliness that makes the game what it is. The story it tries to tell: even the most pathetically weak fish, a Magikarp, and even those Trainers who have to put up with its incompetence at anything other than jumping, have all continued to try and go on and on and on and not give up because of the support from their friends, and you're just one of many people who love Magikarp. There's a whole town of people that revolves around these competitions because everyone, even the pathetically weak Magikarp, has a place in the world where they are loved. There's no way to construe these as gameplay rather than story; they're the game's personality and they're the reason people come back to the game and continue to mess around with it and have fun.
I kinda think all of you are understating story a lot, haha. And I mean, as Aki pointed out, we all have the same gameplay for the most part, and it's pretty difficult to mess it up unless you go out of your way. Besides, maybe nobody is seeking out fangames for their story, but once I've already played it, the story decides whether I tell my friends and recommend it or not - the story still decides if those friends play it. Besides, since different people have different preferences... no matter what your difficulty curve is, you have an audience, so why is that a limiting factor that supersedes and suppresses story? OTL
Then again, I mean, I personally actually liked SM and BW and thought their focus on story made them a heck of a lot more engaging than other games that didn't have story, but apparently that's unpopular because it gets "railroady" - as a slightly tangential topic of discussion, instead of trying to pick one or the other as more important, how can you keep an engaging story that is present throughout your game from becoming railroady and intrusive? To those of you who prefer games with more focus on gameplay and dislike when story gets in the way, what alternatives, changes, pieces of advice do you have to allow gameplay to still shine through and meet your standards without going the other way and getting in the way of the story?