There seems to be a big disconnect in his thinking since this is the same father who said that he is only willing to pay $10k, half the cost of attending in-state. I know that he knows that Yale is more than 10K. Taking in to consideration his and your mom’s income/assets, you would be full pay at Yale. Most top schools that meet 100% demonstrated need will require the income and assets of both of your parents.You can only borrow $5500 as a freshman. You already know that your mom cannot afford to pay much (or at least you don’t want to burden her with paying much). as previously stated, dad will only pay $10k.
You really need to chase merit at places where you can get full tuition or close to it. This way, your dad’s 10k, your $5500 loan and whatever your mom can contribute will cover room/board and misc expenses.
You can consider applying to UChicago, which is a FAFSA only school and Vanderbilt, which will only look at the income/assets of your custodial parent. If your EFC is $35,000 and and your dad pays $10k, you borrow your $5500, will can your mom afford the 20k?