This might be controversial, but I like the idea of school specific exams for elite schools. This allows the school to select the students they want and filter

This might be controversial, but I like the idea of school specific exams for elite schools. This allows the school to select the students they want and filter out the rest. They should have one for each of the majors (perhaps a more general one for related majors). They could just accept the highest scoring students. If there is a tie then they could either perform a waitlist or a random selection.