If we were talking strictly design (as in art) your point would hold a little more weight with me, but we are talking about web design. The skills that are required here are not that different from the skills that are required by a programmer. One is very visual it is true, but both require strong troubleshooting skills.
I'm not sure about the whole left/right brain thing. I'm speaking from my experience of what it takes to be a good web designer.
Is programming strictly a right brain activity? I hear a lot of talk about beauty and design in programming circles today. Clearly it is not all cold crisp logic. You aspire towards beauty and function just as much as designers do.