$180,000 and under, all you have to pay is 10% of your parent's income for Harvard. Above that it scales, but still you have to be well in the $200s+ before you have to pay the full admission price. Similar for Princeton/Yale other schools with a shitload of money, though I'm not sure of the exact numbers.

Given a median household income of ~$50k in the US, I'd say yes, elite schools are almost unequivocally cheaper across the board than state schools. Common misconception held by most.