In academics, you need a DrPH, or PhD, in a related field. In the private world, having a DrPH helps. It is always advised to never enroll in a PhD program which charges their students, for both personal and professional reasons. I believe that BU is one of the few schools that charges for some of the PhD programs, and they just won't get the talent pool that other schools get.
If you want policy/leadership, I would consider a PhD/DrPH at a place like JHU, Harvard, UNC or Columbia.
Every school charges for their graduate programs, regardless of whether or not it's a PhD or not. Whether a student can secure funding to cover those expenses is entirely up to that student. At the time of admission, students can receive funding offers of various sizes (nothing, tuition only, tuition + stipend, etc.) and it's up to the student to decide if that's the right choice. In general, I would say that a doctorate is the right choice only if that student needs that degree for his/her career development. Not many jobs require a PhD and there are fewer jobs out there designed for someone with a PhD, so only get one if necessary. Becoming 'overqualified' is very quick once you have a PhD.
I know for a fact that many students at BU have an arrangement that is similar to UW in how PhD students receive funding. Most of them will work under a PI as a RA and in exchange, receive a stipend + tuition remission/waiver. So wherever you heard that PhD students don't receive funding is completely false.
The only school I know that doesn't charge their PhD students a dime is Yale. Every PhD student at Yale is 100% funded (in every department). One of the reasons Yale's graduate school has a much smaller enrollment than many competing Universities that do offer partial funded and non-funded positions.
While a MPH is a common degree, it's a common degree because it's versatile and teaches many skills which are useful in the work force. There are lots of jobs designed for someone with a MPH so finding those jobs is key. Having a PhD is a way to stand out--but it will exclude you from jobs that require only a MPH (i.e. overqualified). You'll stand out, but perhaps not in the way you think you might. Specific types of jobs need a PhD/DrPH: becoming an academic PI, research director, research associate (title varies by firm), technical specialist (consultant-ish title), and other specialized and technical positions which require expertise in a narrow area.
As it stands, funding for PhDs is very very very tight right now. It goes all way from NIH down to undergrads. It's tough to get in and even tougher to get funding. But if you really want a PhD, go for it. I loved the process of getting one, but I have the feeling that I probably wouldn't get in now if I tried to apply because it's become much more competitive in the years since I was a student.