Game Design should probably be an art school discipline.
When I was in uni CS majors could take multiple gamedev classes but they were electives
BTW my No.1 pet peeve about universities is they force you to take many irrelevant courses which you can simply look up by yourselves if you so wish.
Game development is all about project management, including managing external teams. The CS is just one part that goes into the whole business. You instead will be working with artists and their tools, from exporting 3D models to playing sound correctly. Getting everybody working productively is Project Management 101. As a corollary, having students make a very simple game (like Frogger or Space Invaders) with multiple people, including non-technical people, is great practice for real-world development while not exploding in scope.
"Game Design" is notoriously vague, but college is a unique place where you can get all the building blocks. You really want to have some experience with music, art, philosophy, theater, creative writing, project/business management, and much more. The broader your experience the more it serves your foundation. College is also a great place to meet many different people (artists, business, theater, music, etc) that can contribute and bounce ideas. You will learn about managing people!
Game development is more Art than Computer Science. Yes it is a simulation that is programmed on a computer, but outside low level game engine development you will be working on much more fuzzy concepts like artwork and "fun" far more than the CS heavy parts. Because it takes so many different experiences combined together, it's hard to make a specific elective or specific major. I do think other majors, like Film and Theater, are a good prototype of what a Game Development major might look like because they too have to combine many different technical specialties into a production.
tl;dr: Game development isn't a single major, but a whole bunch of majors in a trench coat.