Perhaps this is the case in sports also, which are not games according to game theory.

[...]

I need to check up on the definitions, but yeah, they make a distinction there.

I think you may have confused something -- perhaps mixing up a distinction used in some specific example with the actual methods of Game Theory, or mixing up game theory with game studies.

Far from specifying a distinction between games and sports, Game Theory isn't even actually strictly about games in the same sense that we're discussing them -- it can often be used to analyse or reason about games, but it's an abstract mathematical system more used for completely unrelated topics such as economics, biology, or political studies, and when used to reason about games is just as applicable to "sports" as to any other game as long as the requirements are otherwise met.