I hate to break it to you, but the last thing that big-name game developers need is amateur or semi-amateur game designers.
If they want game designers, then they can easily advertise for people with both experience in game design (so they can see examples of previously created successful games which the person has designed) and relevant qualifications (such as a degree in Game studies and development).
Many people with both relevant qualifications and experience are being turned down for such jobs however, as there simply aren't the same number of people required to design a game usually, as would be needed to say do programming labour, or slog through a section of a game repeatedly testing it for glitches.
The normal process would be to get qualified as a game developer/designer, then begin as a bug-tester before working up the ranks within a company from there.