That's an elaborate analysis of the considerations of getting a US Ph.D.

Gee, I got a Ph.D. in applied math; it could have been called computer science. But I never went through even 10% of that analysis. I didn't get accepted to Stanford, Berkeley, CMU, or MIT, but I didn't apply to any of them. But I did get accepted to Indiana University, Cornell, Princeton, Brown, and Johns Hopkins and went to one of those.

Why did I get accepted? In my application I showed that I had (a) a good ugrad pure math education, (b) a good career in applied math and computing with some good work results (samples), and (c) a lot of learning of applied math and computing since my ugrad education.

While I was a grad student, I regarded the courses as only vague suggestions. If I found the courses not well done or not much relevant to my interests, then I felt free just to ignore the courses. In the end, I had four courses that were good, one of which I already knew maybe 80% of the material (it totally pissed off the prof that I already knew so much of the material and effortlessly and unintentionally led the class on all the grading by wide margins). The other three courses were good and I learned some good stuff well.

Next were the Ph.D. qualifying exams: There were five exams, and I led the class on four of them. For three of the four, I led because of what I knew before I entered the program.

Next, what really got me on track, waxed the bottoms of my skis, to get my Ph.D. was that in one of the good courses I saw a problem, saw no solution in the literature, and in two weeks found a solid solution. Later I published the work. Any of the faculty members would have been proud to have published that work.

For the dissertation itself, I identified the problem and found an intuitive solution before entering grad school. Next, independently in my first summer, I converted the intuitive solution to solid math. Later in a few weeks I wrote and ran software that did well illustrating that my solution worked, was practical.

So, summary lesson: (i) Arrive well prepared, from ugrad school, work experience, independent study, whatever. (ii) As soon as possible, hopefully before entering grad school, independently get a research problem and make good progress on it. (iii) While in grad school, likely largely independently, hopefully find some little problem, find a solution, and publish that. (iv) Likely independently, finish the dissertation project, submit the written document, and stand for an oral exam. (v) Do expect never to pay any tuition, but don't expect any pay. Instead, have some money, a car, clothes, etc. from, say, savings from work before grad school. (vi) Don't expect much from the department -- don't expect good research projects, good hints for project solutions, etc. Instead, just work and produce good results almost completely independently.

One more piece of advice: For a Ph.D. in any of the STEM fields, start with a good background in pure and applied math, say, foundations, abstract algebra, linear algebra, and analysis. In particular, learn how to write math. Then outside of pure math departments, will likely be the local star in abilities in math. Then for research, mathematize some part of the STEM field.