“The differentiator is what school you go to. It’s the difference between starting say at 100K per year versus $60K a year.”
That is absolutely not true, at least in the way you think it is. They may look at schools but to them, a public flagship is more desirable than a more selective college (outside of the three you mention), for a number of reasons. And the school rarely determines how much your starting salary is, it’s based on how you interview and the budget the dept has that’s doing the hiring.