If you already accept the concept of expected utility maximization, then you could also use mixed strategies to get the convexity-like assumption (but that is not useful if the point is to motivate using probabilities and expected utility maximization).
That is indeed what I had in mind when I said we’d need another couple sentences to argue that the agent maximizes expected utility under the distribution. It is less circular than it might seem at first glance, because two importantly different kinds of probabilities are involved: uncertainty over the environment (which is what we’re deriving), and uncertainty over the agent’s own actions arising from mixed strategies.
That is indeed what I had in mind when I said we’d need another couple sentences to argue that the agent maximizes expected utility under the distribution. It is less circular than it might seem at first glance, because two importantly different kinds of probabilities are involved: uncertainty over the environment (which is what we’re deriving), and uncertainty over the agent’s own actions arising from mixed strategies.