Visual artists are rarely game designers--why?

A couple years ago I ran across an English university that advertised a video game design degree, but I quickly discovered that only one or two of the classes were in game design, and the rest in art and 3-D modeling. The school claimed that this was a program that would lead to game design opportunities. My reaction was, first, I can't think of anybody who comes from the art side to be a game designer although I'm sure there must be some, and second, that "game design" is a "sexy" term that some schools use even though they're not intending to teach game design. I have often seen that. And a later reaction was (in a Rodney Dangerfield voice), "game design don't get no respect" as a separate set of skills.

Recently, I was thinking about why so few game designers seem to come from the art side. I suppose there must be some, but most come from either programming or from writing or producing or working in other miscellaneous functions such as QA, in other words neither from art nor programming.

And I realized that game design is largely left brained rather than right brained, more akin to programming than to art. Game design is about critical thinking and about problem solving. In a very large sense game design is a game in itself, a strategic game, not a "haha" game or a beer and pretzels game or an interactive puzzle. Yes there is creativity involved, but it is mostly a creativity related to problem solving where people are a big component of the "problem", not a creativity related to art. Now I'm not saying that games are not art, don't get me wrong, because I think it's clear that they are art, even though the players don't care.

Having said that, one might suppose that some of the games that are the sole design of the designer(s), that become very successful, for example Pictionary or Blokus, may derive more from an art mentality than the successful games that are designed by people who design lots of games. Maybe.