Thank you for sharing your updated list. Others have already commented on the affordability of the UCs, but what is it that attracts you to those three campuses? There may be schools that have those elements that would be willing to meet your family’s financial need.
Has your family run any Net Price Calculators (NPC) yet? I would suggest they run them at Cornell, Rice, and Case Western. After using the calculator, the school will provide its Expected Family Contribution (EFC). Is the EFC these schools are providing a price that your family is willing and able to pay? If not (and for many people the answer is “not”) then you will need to find a school that offers merit aid in addition to financial (need-based) aid.