Right. It's briefly introduced in geometry (or used to be) in the typical US high school curriculum, but not thoroughly and not as a distinct concept from geometry. It deserves its own sequence of material, and it should be applied to multiple areas so students can see the value of it more easily.

When the only application is geometry, and they hate geometry, they won't care about the "tool" being used.

Aristotelian logic can and should be taught at the elementary school level. Bring back the Socratic method while we're at it! And Euclid's Elements! The Greeks contributed well to education.