I’m not sure where you get the idea that all universities are competitive because that is simply untrue. Many public universities are required by state legislatures to broadly admit students from their own state. California is a good example. You can see incoming class statistics here: https://www.collegetransitions.com/dataverse/entering-class-...

Competitive universities are competitive. Sometimes that competiton is justified and sometimes it is unjustifiably based on pedigree. The program I attended before getting my PhD at a public university was at a well-known private university. But in terms of research productivity and impact, the public university was a MUCH better place to go to school. The PhD program was exceptionally competitive but its undergrad program was not. As a result, many undergrads who wandered through our lab door got a first-class research experience.

I’m not going to pretend that pedigree doesn’t matter (to some people), but it is emphatically not the same thing as “a good preparation.” I took a couple math classes one summer at a low-tier state college. They were just as good as any math classes I took at any of the other places I went to school, public or private.