Number one, there are different types of privates and publics - those designations only pertain to where most of the school’s money and control comes from, not the quality of the school. The UCs themselves are some of the best universities in the country irrespective of public status, with some (Berkeley and UCLA, particularly) being on par with the very elite private schools. Conversely there are many private schools that aren’t really a better choice than a public. Cal Lutheran, for example, is a good school, but I certainly wouldn’t choose it over Cal Poly SLO or UCSD et al. unless it was significantly cheaper.
Some employers care about the undergrad school. Many don’t.
Number two, I do sometimes think that undergrads overestimate the importance of the networking/alumni connections and whether they think that will get you a job. On-campus recruiting is tremendously important in some fields. In others, it barely registers. Alumni connections and career center machinations can be useful in some areas but there are lots of other ways that graduating college seniors find employment.