As a appendixe or the like for a game design doc would it be a good idea to tell a imagionary game as a story...... to set the mood for the team members

That comes down to what your game is.

If the game is mostly a story vehicle, you will probably end up summarizing the story.

If it doesn''t offer a varied enough palette of "game situations" then you will only need a few lines to cover the whole range of gameplay and won''t have to detail example situations, which is fine.

If the above two don''t really apply then you probably can set to work making a "mini-story" out of your imagined gameplay and it''ll be reasonably helpful, though it could divert focus in the wrong direction by highlighting some aspects over others.