Can who you pick as "faculty members of interest" make-or-break your PhD application?
The answer depends on the admissions process at the school.

Some departments will look through all applicants together, choose the top applicants from the entire pool, and then assign them to faculty advisors. Student and faculty preferences on the advisor has some impact, but these students are competing against all other students.
For other departments, individual faculty primarily look through students who want to work with them (as stated on their application), and admission decisions are made for individual research groups. For these departments, the preferred faculty advisor that you list can have a large impact on the extent to which different faculty review your application, and which other applicants you are primarily being compared with.

As an externally funded PhD student, you are not competing for the limited funding like other applicants. Of course, there may be other constraints such as lab space and available time for specific professors, but available funding is often one of the main active constraints. That gives you a greater chance of being able to work with a specific advisor.
You will likely be working with the same professor for 3-4 years, so requesting an advisor whose research interests and/or working style are a good fit for you could make for more enjoyable and productive PhD studies.