Game design is in fact more about daydreaming and spewing ideas.
"Daydreaming and spewing ideas" is the activity of arm-chair designers and the dreaded "idea guy", but it's only a small part of what real designers (that is, professional game designers or those who successfully design for indie or hobbyist games) do, and I would tend to agree with Legendre's suggestion that a lot of the difference is in making your ideas really work -- and work well -- within a set of constraints. See:
- Sloperama lesson #14: All about the job of "game designer"
- Extra Credits: So you want to be a game designer?
- Evolutionary Design: A practical process for creating great game designs.
- Computer Game Development: An Overview
If you're happy to be and remain an arm-chair designer that's fine, but the job done by those who professionally work in the role is more about implementation and less about day-dreaming and thinking about ideas. Anyone can daydream and think of ideas -- a designer must be able to take those ideas and make them work within a given set of limitations.
Do you want to be a real designer, or just someone who thinks of (and writes down) ideas? Either is a fine goal if you're happy with where you're at, but don't think more highly of your role than the reality -- if you want to take the step from idea guy to designer you have more work to do.
However, you don't need to retrace all the fundamentals of rocket science (and experiment with what has been done before) to advance in the field.
Actually, that's exactly the material that some of the introductory courses in an aero-space degree (or related courses) cover, and it's common in many fields to cover at least briefly the work that has been done before. In any technically complex field you simply cannot -- savantism aside -- advance to the bleeding edge current research without first having a sound practical understanding of the fundamentals.
Architecture students begin by studying existing structures and designing smaller, less complex buildings -- they don't go straight to a sky-scraper.
Physics students begin by studying classical mechanics, dropping balls and rolling things down inclines to learn about gravity, friction, etc. -- they don't skip to solving the mysteries of the universe.
Game developers begin by making small, functional games or prototypes -- they don't jump straight into complete large AAA -- or even good indie-quality -- titles.