So, if it's any comfort, this is a competence-based field. There are some exceptions to the rule, but where you go to school is much less important than your work ethic, intelligence, and competence. And where you land out of grad school is much more important than where you land for grad school.
Don't believe me? I worked on Wall Street for five years with a state school on my resume. It would have worked against me if I were working on investment banking deals, but I was a good fit for quant work. As long as you learn the material, branding is much less important for quants than it is for most other people.
So, if it's possible, try not to get too worked up about the outcome. Your future self has too much to worry about with building volatility surfaces or building a support vector machine learning program to generate some pricing signal to be worried about which school you went to. It may feel like cold comfort now, but in five years, you'll agree with my claim that where you go to grad school does not matter that much if you want to be a quant.