Best interview I ever had was with a Hispanic lady that had just graduated from a state school (not one of the premier schools). She came out of the LA ghetto, worked her way through college and was the first person from anywhere in her family to graduated from college. Her “never say die” attitude was unbelievable, I knew this gal was going places. I hired her and I was right. Plus she was a nice person.
You absolutely do not have to go to an elite school to be an elite pro. In fact, most of the elite pros I know went to “ordinary” schools. Heck, even the technically elite MIT guys are all brilliant but frankly none of them I work with are likely going to climb high because they lack polish in presentation. That is, they tend to have a “isn’t this cool, check out this data” presentation style that confounds managers and customers. Hard and soft skills together make the leader, and schools may teach one or the other but most don’t teach both.