Why is classical geometry usually not taught in universities?

I don't know anything about the Olympiad or Putnam world (at least not since my own undergrad days a long long time ago). But I'll answer the title question from a different point of view.

I think there's still quite a lot that a low-to-mid level undergraduate geometry class can still offer even to students who intend to pursue a graduate degree in mathematics.

One way to teach such a class (alluded in the comment of @Mason) is to emphasize the 2000 year quest to resolve whether the "parallel postulate" is independent of the other axioms. This quest culminated in the 19th discovery of hyperbolic geometry. Knowledge of the three basic 2-dimensional geometries --- Euclidean, hyperbolic spherical --- is an essential first example of the modern approach to classifying geometries.

Classical Euclidean geometry does not have to be taught in the classical method. For example, there are books in Euclidean geomety which argue that Euclid's book carefully avoids symmetry arguments and therefore their book is going to avoid symmetry arguments in their book. I find this to be an unfortunate pedagogical choice. Instead, another way that I teach axiomatic Euclidean geometry (which remains an excellent introduction to axiomatic methods in mathematics) is by a method which embraces symmetry, building it into the axiomatic approach. I then go on to build up to the classification of the 17 wallpaper groups, which is a great way to introduce some group theory. In fact, in my upcoming graduate course in geometric group theory, I intend to give a summary of the whole semester of that undergrad course, including the classification of wallpaper groups, before moving on to more modern geometric classification theorems.