Well, I work from the assumption here that you already know that Game Designer =/= Game Developer. And I also want to tell you all you get from me is second hand information, as I neither are a game designer nor professionally work in the industry....
Game Designer is a Specialization, AFAIK best compared to the director of a movie. As such, again AFAIK, you will need SOME competency in ALL the specializations involved in creating a game.
You will have to talk to the programmers, the artists, the sound engineers and artists, story writers and of course, the suits. So you will need to understand their language, and to some extend what they do all day long.
As such I would say, yes, a CS Study could help you with your goal of becoming a Game Designer in 2 ways: you will already be proficient in one of the specializations you will have to work with to create a game. And it is as far as I heard one of the easiest ways to get into a game studio (easy as in: you will find a job easier, because there is high demand and less applicants).
You might even be asked as game designer to do some lightweight scripting in some studios it seems.
You need to be aware though that the main job of a game designer is different from a programmers job. If you want to know more about that, best head over to Tom Slopers page at http://www.sloperama.com and look at his all encompassing Q&A for aspiring game designers... good read, not only for people interested in becoming game designers.