Can game theory be used to predict economic specialization (division of labor)?
I think you're over-analyzing here. Game theory is not just about maximizing your own pay-off, but about doing the best you can given the behavior of others. What Smith describes is just that people do what they're good at to make a living. That is, the way I read it, Smith claims that people specialize independent of what others do.

That being said if the emergence of specialization was to be predicted with game theory I'd look into evolutionary game theory. I'm not sure if you'll find anything though.