That assumes that the model is assigning vanishingly small weights to truly incorrect answers, which doesn't necessarily hold up in practice. So I think "Unless you screw something" is doing a lot of work there

I think a more correct explanation would be that increasing temperature doesn't necessarily increase the probability of a truly incorrect answer proportionately to the temperature increase (because the same correct answer could be represented by many different sequences of tokens), but if the model assigns a non-zero value to any incorrect output after applying softmax (which it most likely does), increasing the temperature does increase the probability of that incorrect output being returned.