As some commenters have mentioned, we should be able to derive lessons of rationality from any game (such as Backgammon or Chess), not just Go. After all, rationalists should win, so whatever helps us be rational will help as win, at games as at life.

However, just because being rational helps with the game, that doesn't mean that students of the game will learn this. (After all, being rational helps with real life, yet many students of real life miss these lessons entirely!) Each game (at least each that has long been widely played) has its own literature and tradition of strategy with maxims and advice, and this is what students will learn.

Now, Go is so simple in principle that one quickly gets past advice which is specific to the game and into advice that applies more widely. So we actually have literature, written by specialists in a game rather than in rationality, that stresses (for example) the importance of accurate beliefs. And so players actually learn this lesson.

(To see if Go is really special in this regard, we'd have to actually check the literature of other games. Don't trust my just-so story about how Go's rules are simple, etc; that's unsupported twaddle.)