Why are PhD programs in statistics coursework heavy relative to PhD programs in the basic sciences? This might go better on academia stack exchange, but it seems probably the people who know the answer are on cross validated..
This is a pattern across the US.  My classmates in, e.g., basic science PhD programs take only a few courses. The structure of the PhD is far more like an apprenticeship. Conversely, my classmates in statistics PhD programs take on the order of 12 courses.  (Computer science PhDs take about the same.)  Why the difference?
 A: I don't think the coursework is intended to be there as busywork in case you prove the Riemann hypothesis on your first day.  More likely, the faculty has made the decision that it wants to get all of its graduate students up to some minimal level of mathematical/statistical competence prior to research training, and a substantial program of graduate-level coursework achieves this.
University faculties have a great deal of discretion in deciding on the coursework component (if any) of their PhD program.  Some have no coursework, and some have a substantial amount of coursework.  In cases where a student comes in well-prepared (e.g., with an existing coursework Masters), the faculty might exempt them from some or all of the coursework.  These decisions tend to be made at the level of each faculty, so  they depend heavily on the preferences of the Head of School, the Graduate Coordinator, and other senior academics in the faculty.
I can't speak to what is common in the US, but when I did my PhD (Statistics) at ANU (Australia) there was no coursework in the degree; all the entrants had either done a full year of Honours-level courses as undergraduates, or a Masters degree, or they had substantial industry experience, so they came in with a fair bit of solid coursework behind them already.  Evidently, in that particular case, the senior staff in the faculty decided they did not need us to have any more coursework before starting research training.
A: That's a very country specific thing and I suspect that it did just grow historically. Many other countries are much more focussed on doing research leading to publications and/or a monograph. This mirrors differences that you can also see in undergraduate education (application first vs. "thou shalt not touch a real dataset without deriving measure theory from first axioms").
E.g. for my doctorate in mathematics (mathematics due to the department, but really statistics) in Germany I attended no courses (some universities in Germany have some required courses), but some events at which people (including myself at the start of my research and prior to the viva) presented their research. Instead, I published two papers, wrote a single booklet based on these (and my other results) and defended a viva.
One disadvantage of the solely research focussed approach is that there are few intermediate goals on the way towards your final thesis and viva (thesis defense).  Additionally, the duration of the program is often less clear. Arguments I have heard for it is that it teaches independent research and that course work oriented PhD programs are more like more-in-depth master programs.
However, I am not aware of any research/data that really shows that one approach is better than the other in terms of e.g. graduation rate, amount/quality of subsequent research output or success on the job market.
