To answer my own question, I’d say it depends on the model, and the stringency of the assumptions, and whether or not the model’s (and the economist’s) focus is on understanding human action, or on constructing an elegant, self-contained model.
I agree with Tyler Cowen’s list of ways that economics research can derive testable predictions from game-theoretic models. I think a lot of the interesting and valuable thinking is in the institutions (there’s a surprise!), because game-theoretic models that are most likely to have some explanatory teeth are likely to have multiple equilibria. Institutions are a way to explain how we end up at one equilibrium instead of another. Behavioral, evolutionary, and experimental research all deal with institutions and can shed light on this.
But I do have a snarky follow-up question for Tyler: to what extent do you think that our colleagues are interested in understanding human action? What about the extent to which we economists are enamored with our tools, and engage in “math for math’s sake” work? Just asking …