Sure, but my point is that the computability of something is a poor measure of its complexity as a game of the non computable aspects are game theoretically alw

Sure, but my point is that the computability of something is a poor measure of its complexity as a game of the non computable aspects are game theoretically always a dumb idea. You could make a similar claim about a game where any player can win at any time if they choose, but can alternatively do anything else. It’s not computable, but it is irrelevant to the complexity. You could probably also argue that a Turing intending player is still computable if playing against someone with a defined strategy. Because they will be halted by the other player winning.