Clarifications on American graduate school for an European student
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!
The rationale is because graduate programs in the US are generally structured assuming the incoming student does not have a masters, and got a bachelors in the US, which can sometimes involve an embarrassingly small amount of math (and maybe more importantly, there is a very wide variation in what students learn at different universities, due to the flexibility of the US system). So, you're entering the system at a drastically different level of preparation from most people. Of course, it seems off to you. My experience at quite a number of US schools is that almost all incoming students need some sort of preparation of this type, often even ones with credentials suggesting that they shouldn't. I think some of the logic here is that it's easier to set out somewhat tougher requirements and then make exceptions to let people accelerate than the other way around.
I think it's hard to assess whether it really is. But I don't know anyone who seriously suggests we don't need to do something along these lines. If something is covered in the syllabus of these classes, it is because the students don't consistently come into the program knowing it.
Maybe I'm being too dismissive here, but I am never sympathetic to these complaints. If you really know the subject, the classes will not take much of your time. No one will stop you from talking to professors about research or doing reading on the side while you're taking the classes, and generally the timetables around exams and classes are flexible. That time will only be wasted if you decide you want to waste it.
In US programs I know about, e.g. UC Berkeley, there are preliminary exams (typically taken after the 1st year), on a broad range of topics, and qualifying exams, on the intended area of the PhD. Unless you went to some very, very good MSc programme, this is meant to prepare you for academic career, and broaden your view of the subject.
I wish I went to one of these, instead of diving headfirst into PhD research.
As to "why is it effective", consider excellent products the system has been producing :-)