Story is an important thing in genres where it makes sense -- yeah, if you're totally into arcade button-mashing, it's probably not necessary. If you want an intense experience on a more personal level, then there will need to be a plot to structure your actions around, otherwise your character's going to have no reason to do much of anything.
In reality, though, story isn't exactly just "becoming" important, as if we've been in the dark ages for three decades... On computers, it was important in the appropriate genres until technological advances changed the focus to graphical quality, basically as it had been for the consoles for years, and FPS started becoming the expected norm. I'm not saying that everything was great (there was plenty of garbage then, too) but there were quite a few games that were. Richard Garriott was quoted somewhere (in an article here?) as describing the cycle fairly well here:
"Periodically, whenever there's enough technological advancement from what the games have done before, you see everybody dives into very simple game play, like a straight shooter with no other frills - it just showcases the technological advancement of the day. However, as time goes on, in order to compete with that game...it becomes more important to deepen the product in some way...and you get more diverse offerings -- more of the things I enjoy, more literary content... I actually look forward to the time when you can start using tools you've developed, and the platforms actually begin to stabilize, and we begin to compete not just on the bells and whistles, but on the depth of the content."
(Of course, that's not counting the games designed for when people don't feel like dealing with depth, or the genres where depth is largely pointless.)