There's many opportunities to go through one of these institutions - you could do undergrad there, grad school, or one of multiple post docs. There's lots of people that go state school -> ivy/prestige -> state school and end up professors. Or state school -> state school -> ivy/prestige. Or state school -> state school -> state school -> ivy/prestige. If you work really hard, chances are you end up somewhere prestigious eventually. Probably hard to avoid.

I was also going to say that there are plenty of less prestigious schools that graduate a ton of professors, like UC Berkeley, but it turns out it's on this list hah.