I can say about mathematics. Where you go as an undergrad isn’t as important as grad school. But you obviously need to go to a college with pretty good math department which has advanced courses and get yourself noticed in order to get to a good graduate school. Many state flagships qualify. I don’t know much about Regis.
Landing a tenure-track position is extremely hard and depends more on your advisor and the quality of your research. Many very well-known professors teach outside of the Ivies. It’s of course generally easier to find a good dissertation advisor in a larger and more prestigious department.