"Nash equilibrium" is far from being a generally convincing argument. Mathematical intuition module doesn't come up blank, it gives probabilities of different outcomes, given the present observational and logical uncertainty. When you have probabilities of the other player acting each way depending on how you act, the problem is pretty straightforward (assuming expected utility etc.), and "Nash equilibrium" is no longer a relevant concern. It's when you don't have a mathematical intuition module, don't have probabilities of the other player's actions conditional on your actions, when you need to invent ad-hoc game-theoretic rituals of cognition.