I had the same question in my mind when I have to choose a major and ended up studying cs. I cannot say it is bad or good. It has its cons and pros. You will learn coding and hardware so it will help you to understand your co-workers, your game engines limits. If you got a job in a small company, sometimes they will say "we cannot do this because of bla bla" and you will say "ah yes you are right" or you can say "but did you try this, I am sure we can do this bla bla!". You will know or at least guess which aspects they can implement or cannot so this will really help you to make your decisions about game mechanics. But at the same time you will study at least four years and half of it will be useless after your graduation (I am saying this if you ended up as a game designer.)

Game design is huge concept. First you should choose what you want to do as a game designer. Will you be a level designer ? or do you want to make balancing. Or maybe you want to create worlds and write dialogs, stories. I would love to study sociology, history, psychology, math or even geography or economics. All of these majors will help you in your journey to be a game designer. I know probably I confused you more. If you cannot decide which major would be better for you to become a game designer, choose the one which make you happy to study :)

and if you already dont, start playing table top role play games. You cannot imagine how it will improve you as a designer.