That doesn't have anything to do with my point above, which is that the alternative hypothesis "people with some property X are better at game Y, and separately

That doesn't have anything to do with my point above, which is that the alternative hypothesis "people with some property X are better at game Y, and separately, people with property X are more likely to cheat" is not ruled out by experiment 1 as described. Edit: you could also differentiate between "winning->cheating" and the confounder possibility by having the cheatable game first.