resources that build connections and present far more opportunities during school and post graduation. It is possible to get those benefits by doing a BYU undergrad then a grad ivy if grad school is in the plans, but be aware that elite schools like to take students from other elite schools into their grad programs, so it is harder to get in as a grad student. Not impossible, but harder.
As for cost - most, and maybe all, do not consider finances in offering acceptance and instead will help with grants and scholarships based on financial need that will cover what the student and his/her family cannot.