That maybe true if you are comparing pure numbers, but as reported here by some of the folks and also in the news/media, a lot of the private schools know they are expensive and offer grants/aid (free money) to make the cost comparable.
I forget who, but someone here said they are at an ivy for not much more or even less than a UC school.
Also, for those on the lower income, if Stanford or Harvard is calling, you get to go for free.
With the benefits of private, you don’t have to worry as much (I don’t think) of getting classes (always full and offered once a year at UC engr classes) so if anything, I’m hoping my kids go to a top private school instead of UC (as a UC grad myself)…