Yes, I got a Ph.D. in applied math from a famous
research university and for a while, for reasons
having to do with my wife, was a prof in a well
known MBA program.

Yup, I've seen Ph.D. programs destroy a lot of
really good people. People crushed for life,
suicide, etc.

I got through okay, but some of the politics was
grim. I got into a fight, had to take a year off,
and, net, a department Chair and three profs got
fired.

How'd I get through? Mostly just did the work on my
own. Entered the program very well prepared.
Brought my own Ph.D. dissertation research problem,
with a good intuitive understanding of how to get a
solution, with me to the program and did the real
research part independently in my first summer.

Along the way I polished my halo:

One way was in a course, supposed to be really hard.
The course was carefully graded, and the intention
was that the class be competitive.

But before the course, I'd studied the material, in
part in courses but mostly independently, over and
over from a stack of the best books, elementary,
intermediate, and advanced, applied the material,
understood quite a lot about the corresponding
numerical analysis, had written software for the
material, etc. I could have given all but a few of
the lectures on the first day of the class. So, on
graded homework, tests, mid-term, and final. I blew
away all the other students by wide margins, and I
wasn't even trying to be competitive.

In a course there was a question but no answer. So,
I asked for reading course as a chance to find an
answer. Two weeks later, from working sitting by my
wife on our bed as she watched TV, I had a nice,
clean answer, with more than I'd hoped to get. Two
weeks, course over. Work publishable -- did publish
it later.

So, I suggest:

(1) Be very well prepared, from undergraduate
school, a Masters from another school, on the job
learning, independent study, whatever.

(2) Get an applied Ph.D., say, in engineering or
"applied science* or some such. Then for the
research, start with a problem from outside
academics. Get at least a good intuitive solution
before entering the Ph.D. program.

If the lectures, seminars, etc. of the program can
give you some tools, ideas, etc. to help you with
your research, fine.

(3) Do not ask for a research problem or research
direction from a professor. Instead, just do the
work independently. If there is any question about
the quality of the work, then publish it or at least
get it accepted for publication.

The big question, though, is why bother?

One point: Usually a person without a Ph.D. doesn't
want to work with a person with a Ph.D. The person
without can feel intimidated and threatened, and
that one might guess that a Ph.D. might be relevant
to his work he can take as an insult. People
without have a lot of ways to denigrate people with.

Some of the challenge of a Ph.D. is summarized by D.
Knuth in a remark in his The TeXBook:

"The traditional way is to put off all creative
aspects until the last part of graduate school. For
seventeen or more years, a student is taught
examsmanship, then suddenly after passing enough
exams in graduate school he's told to do something
original."

That "suddenly ... told" can be a big shock.

Here are two common problems:

(1) Good Students.

Ph.D. programs tend to want only good students as
in PBK, Summa Cum Laude, Woodrow Wilson, NSF
Fellowships, etc.

Well, one of the more common ways to be such a good
student is, in addition to being bright, liking the
material, being highly determined, and working hard,
is to be terribly afraid, of criticism, failure,
failing to come up to what parents wanted, what high
school teachers expected, of some relative saying
that they "expect great things", etc.

So, such a student can be a case of anxiety
disease, have done well in K-12 and in college just
by making A grades and otherwise not thinking much
about anything else. Then the "suddenly ... told"
can be a severe challenge, a risk of failure, of the
first criticism in life, a threat to self image as a
great student, etc.

There can be anxiety, stress, depression, failure,
more stress, clinical depression, and suicide.

It can help to have (a) a thick skin and (b) good
grounding academically, etc. before entering a Ph.D.
program.

I got a thick skin in grades 1-8 -- the teachers
treated me like dirt. With my success in math in
grades 9-12, I got a little better treatment. But
the big day was when the SAT scores came back and
the teacher who'd had me in the sixth grade read the
Math score and, gulp, "There, uh, there, uh, must be
some mistake." Yup, there had been, hers and that
of the rest of the teachers, for 12 long, painful,
largely wasteful years where I'd been treated like
dirt.

But someone up there liked me: I got sent to an NSF
summer program in math and physics.

I'd learned that there was no direct way I could
ever hope to please the teachers, but with math I
could do work that neither they nor anyone else
could fault. So, math it was. And physics.

(2) Opaque Criteria.

Mostly students are not told just what the criteria
are for the work, the social aspects of the field,
the department politics, for research, for
publication, for a dissertation, etc.

So, students can do too little in some respects and
get into trouble or do too much in other respects,
go too slowly, and again get into trouble.

My solution: Largely avoid the opaque stuff. For
the learning, do that largely independently. For
the research, do that independently and well enough
to be publishable, e.g., "new, correct, and
significant" and, in case there is any doubt, just
submit the work for objective, blind, expert
peer-review and, then, publication.

I haven't been at all interested in an academic
career or much interested in publishing, but all the
papers I've written as sole author or co-author have
been accepted for publication right away; so, it's
possible to follow the work style I've explained
and get published. Getting published in academics
is much like making money in business -- lots of
other potential problems melt away.

For efforts, work style, such as I have described
here, independently and/or in a grad program, is
there any benefit? Well, all the best work I've
done in my career has been from just that work
style. Currently I'm doing a start-up, and the
crucial, core secret sauce is from just such work,
with some of the grad school work a big source of
the prerequisites. So, if my start-up is
successful, then the work style I explained above
will have been successful.