Clarifications on American graduate school for an European student My advisor strongly encouraged me to check out two (highly renowned) American universities for my PhD study. 

Her collaborators in these two institutions are good people and great mathematicians, so, assuming I can get in, working there would be very pleasant. 

However, I'm confused by the structure of these graduate schools. In the first two years I'm supposed to follow courses and do a comprehensive written exam and an oral one  which are mostly about topics (complex analysis, basic functional analysis, ODEs) that we do in the first two years of undergraduate study in my current institution. 

The exam questions from the past years appear to be quite difficult, but revising those relatively elementary topics and spending a great amount of time solving difficult problems on them  seems like taking a step back after a Bachelor's and a Master's degree (which had a quite significant research component). 

So my questions are the following:


what is the rationale behind this structure of graduate school in the US?
why is it effective?
should I be concerned about "wasting time" revising basic topics in my area instead of diving directly into a research program after a Master's degree?




Added context from comments:

"Students who already know the material can take the exams in the first month of their Ph.D. program" is exactly what my advisor's collaborators told me. However, although the core material is well-known to me, it appears that the exam consists of many problems in a short amount of time and that such problems are mostly about clever ways to sum series, evaluate multiple integrals, do contour integration, solve tricky ODEs, and so on. That is, it is about elementary things but requires lots of exercise. That's why I'm concerned that it could be an unnecessary detour.
Speaking as a long-time graduate program director and adviser in a mid-level US department (at a big public university), I'd emphasize that most of our Ph.D. students in past years have had a less-than-adequate preparation in some basic areas.   This is true especially for US students coming from a typical 4-year college math major but is also true for some international students.   On the other hand, most of our students aim for a college teaching career, or perhaps a non-academic job, rather than aiming for a research career in some kind of mathematics (or statistics, part of our department).   At the most elite departments, it is taken for granted that everybody plans to do research.    But my own contemporaries in the 1960s Ph.D program at Yale actually went in many different directions.    

Like most other US departments, we have had an evolving procedure involving written (and perhaps oral) qualifying exams.    Anyone applying to one of the top few programs should certainly ask questions about how and when such exams are administered, and how they are evaluated. Typically no one wants to hold back a talented student, but sometimes a student overestimates his or her own talent and knowledge (and future job prospects).    Even Harvard and MIT Ph.D.'s sometimes end up teaching at small colleges or out-of-the-way universities.

In any case, identifying a potential thesis adviser (or two) is equally important, though obviously it's difficult to predict one's future interests precisely or to predict the future logistics of an active faculty member (sabbaticals and other leaves can upset plans as can personal crises).   Good luck navigating the US system, including the evolving immigration rules!