I've also heard that the US has a very good system of higher education compared with other nations, though it's almost universally understood that our system of primary education is the pits.

Yes and no — post-graduate education in the US is excellent but the educational pit is also swallowing the four-year university. The percentage of foreign students in US universities skyrockets at post-graduate level in part because US undergraduate programs are doing so poorly, relatively, at preparing students to get into US grad schools. (Or at least this is what I've heard. I don't have sources ready at hand.)