Top private schools are often effectively cheaper than UCLA for most people. For instance, Stanford waives tuition for students from households in the US with income in the bottom 80% (under $100k). For students from households in the bottom 50% or so, they also waive room and board.
Which is ironic? is maybe the word? given how difficult it is for people from families in the bottom 50% on income earners to get into Stanford; that's something like 15% of the each incoming class. Admissions at Stanford is need-blind, but many important college admissions criteria are proxies for need.