You fixed the rules of the game. It follows that there is a distribution of outcomes of the game (it was guaranteed to end). This distribution is not about time

You fixed the rules of the game. It follows that there is a distribution of outcomes of the game (it was guaranteed to end). This distribution is not about time-averaging or ensemble-averaging. There is no time, there is no ensemble. Don’t you agree that there is a probability distribution that describes the outcomes of the game you proposed? If you don’t, that’s where you’re mistaken. Expected utility theory uses that probability distribution of outcomes. In this case it was directly derived from your description of the game. The probability distribution could also represent the probability assigned to the different potential outcomes by the agent, and another agent could have a different probability distribution. In any case, there is no notion of ensemble average over N realizations or time average over N realizations or anything like that in EUT. There is a set of possible outcomes, which have a value called utility attached (representing preferences). For each possible decision, there is a probability distribution over the outcomes. An expected value can be calculated. The optimal decision is the one with highest expectation of the utility. Saying that utility theory has an issue because of typical trajectories and time averages and ensemble averages doesn’t make much sense. All those things are not part of EUT. You could just as well say that it has a problem with the spark plugs. If you say that you may solve some problems differently, that’s ok. That doesn’t make EUT wrong or disallows the use of probability distributions to calculate expected values. And as far as I know, all the decision problems solved by EE have a straightforward correspondence to a EUT solution with some utility function (and with a perfectly standard probability distribution that can be used to calculate expected utilities).