Most internships are not with expensive law firms, politicians, or banking companies. Most are with corporations, especially large tech companies. The people who are in charge there generally are not graduates of small private colleges, but of large public engineering powerhouses.
Access to internships at tech companies is, for the most part, fairly straight forward (apply, code test, etc) so students from any school can engage (even without on-campus recruiting events). Campus recruiting events do, however, make it easier as some hold first round reviews and even code tests on-campus (advantage schools with many of these). These events are typically listed on department websites.
There are, however, many ways that networks can assist. For instance, at many tech companies you can skip the code test and go directly to f2f interviews if an existing employee will vouch for you. So, at schools whose graduate programs funnel into specific companies the coattails of the grad students assist the undergrads (same when profs also work/consult at certain companies). Another way networks help is in startups where current students (grad and undergrad) as well as profs either hire interns or help them engage with startups (the startup culture is heavily word of mouth).
Another reason a California kid may not want to go to a UC is the quarter system. If the student wants a semester system, a private school in California may offer that
All of these outliers don’t answer why such a vast majority of the top half of the class choose private.