[hopler]

Princeton math (and other?) department specifically advertise that their research-only PhD program is unusual or unique.

https://www.math.princeton.edu/graduate

Generals/Quals require 1-2years of graduate level study, not remedial undergrad, if you look at the material covered and the syllabus and levels assigned to courses at various colleges. It just so happens that at a school like Princeton, 1st year PhD students have already taken graduate level classes in their field, or equivalent self study.