No he didn't. He compared Harvard to Berkeley. Berkeley is another elite institution. The cost of attending a university depends on its status, not whether it's public or private.
What leads to your belief that about the cost of universities? The costs are quite opaque.
Private schools all have the same nominal tuition. The most elite ones have the biggest endowments and appear to me to give more financial aid. Less elite ones do try to lure students away from elite schools with merit grants, but I think that merely allows them to match the price of elite schools for the very few students that they are able to lure away, not undercut the price.
[Added: Solipsist made good comments that partially account for the phenomenon, and Douglas_Knight pointed out that the figure of $200m+ below should be $30m+]
This article reports that Harvard has 2,964 alumni worth $200+ million, with a total wealth of $622 billion. These figures are staggering:
Note that Bill Gates "only" has ~$75 billion and that Mark Zuckerberg "only" has ~$30 billion, so that they don't account for Harvard's decisive advantage over other universities.
What is going on here? Why would Harvard come out ahead by such a large margin? Its acceptance rate is smaller than those of Stanford, Yale and Princeton, but by no more than 25%. Moreover, this has been true historically
One can ask a similar question of University of Pennsylvania, which is significantly less selective than Yale and Princeton, but has a decisive advantage over them.
Some possible explanations for the discrepancy:
Regardless of which of these explanations hold, there remains a question of why they would hold.
Is Harvard a better choice than other universities for students who aspire to be wealthy than other top schools?