BMHEE: Family size counts best if there are multiple undergraduates (including college age yeshiva and seminary students), but it still counts. Even a public schools, but especially at private schools, the FAFSA’s EFC is the starting point. One needs to haggle. Unless you are a super-rich legacy admission, or poor enough to get an EFC of zero (living off of tsadaka), you need to haggle.
Also remember that the more overqualified you are the better your chance of a merit scholarship. The person who would be full tuition at Columbia or NYU, would warrant a merit scholarship at a lesser school (they need good students to boost their ratings). Also remember that the elite schools come close to or meet the EFC figures (meaning they give aid to cover everything beyond the EFC, something not true of lesser schools). Unfortunately, YU (which is going bankrupt) is a lesser school in this respect.