Is a well-written story important for me in games?
My answer is no, absolutely not. Because some games are great because they ditch story and just go for sheer fun. Others are nice because they're very heavy on the story.
What is important is that the game has something, some element that doesn't have to be story, that makes it special. Chrono Trigger is super-special because of its epic story and awesome music. Gameplay-wise it was what is now considered standard final fantasy fare, so that's not one of the reasons why one would play that game.
On the other hand something like Torchlight. It's got a story but you can't call it a deep, well-thought-out one. However its gameplay is excellent; it takes the Diablo format and makes it deeper and more enjoyable.
Oblivion and Fallout 3: The stories in those games aren't all that spectacular either, and gameplay has plenty of flaws, but Bethesda is the only bunch in the world who make games like that, and both have that unique Bethesda touch.
No, story isn't important to me. It can make a game great, but it's not the only thing that can make a game great. If an epic story that was written with a level of mastery you'd find only once a decade was an absolute necessity, I'd be an EXTREMELY dissatisfied and bored person.
And to go further, story doesn't have to be extremely complex, character-driven epics either. Some stories are great because they aim for the mindless side of things, and often they excel so much that they become greater and more enjoyable than some deep, story-driven stuff.
I'm sure many have seen Guu. Not much of a story, but it is AWESOME(tm) because it knows what it's aim is, what its strength is, and makes all effort to convey and capitalize on them and hardly try to venture into areas that detract from the virtues of that particular story.