I know this might come across as kind of a shock to some of you, but I'm going to have to say it anyway. Not all games are the same. I say this in light of having just read Ian Bogost's article on conservative and aberrant reformists in game design (and the aesthetic filter). But really, that's just the trigger for this post. I've been talking about this for months, although my fervor was slightly diminished when I found that Jim Gee had already touched on the topic.
At any rate, I'm trying to make a more pragmatic point than Ian does, and I also feel a need to bring more complexity to it than Jim did (although most of the complexity will wait for another post). Basically there are many different types of games, and they require many different types of designers.
Much as different sorts of print text require different sorts of expertise to produce, different types of games also require different skills. Hence, when it comes to discussing "what games are for" or "who should design games", I ask you to remember that there are lots of different kinds of games. For the designer question, the important thing is to make sure you have the right people on your team to make the kind of game you're developing. As far as the purpose of games, there are different games for different players. That should be answer enough. Dan Cook might claim that narrative games are an anomaly, but I'm willing to guess that there are a whole lot of BioWare fans out there who would be inclined to disagree with him.
One final point to make and I'm done. I'm not just talking about genre here, I'm talking about form. Poem vs. novel vs. exposé, not horror vs. fantasy vs. historical drama. In games, we often confuse genre and form, but that's a post for another time.