I am taking a course on game writing and learning the hard way how a game can live and be successful without a story. Games do not need stories.
When gameplay and story marry well the story can boost the experience for it be remarkable and memorable. This is because play is a problem-solving activity, useful to improve some of the skills useful to survive. Stories instead help us understand better people and how to relate with them. Games are about things, stories are about people. This is why is so hard to link a game with a good story.
Point of touch among design and writing is that designers look for fun. And fun is a feeling. Writers look for feelings. You can easily spot there is an interesting overlap.