Really? I would naively expect the outcome to matter a lot more. As a PhD, your work product has a lot more transparency than B.S./M.S. degrees, because people will actually hear about your work if it matters (for several different, valid definitions of "hear about" and "matters"). However, I would also naively expect the name of the school to correlate with outcome, but only because because different professors work at different schools.
The outcome is severely affected by what the faculty are doing at the school you end up at. Your chances of successfully completing a Ph.D. doing something completely different from what people are doing in your department are slim.
