The setting of the story in Afghanistan is pretty vital to the plot. It would certainly be possible to tell the story in another place: there are places that go through upheaval and civil war and there are places where there are oppressed minorities. But given Hosseini's experience and knowledge of place being tied to Afghanistan, it is difficult to imagine the novel being well set anywhere else.
It is important that the action of the play be set in this astonishingly beautiful place, particularly because so many readers come to the novel with a different impression of Afghanistan. Against this backdrop, the friendship and then betrayal between Amir and Hassan is powerful and moving. And if the plot was going to be built around the Kite fighting competition, there's no place where it is as beloved and wide-spread as Afghanistan so this obviously plays an integral role.