Perhaps we could have a nationwide entrance exam acceptable to most or all universities. It would test the students' preparation for and likelihood to succeed in achieving a Bachelor's Degree.
We might call it the Scholastic Aptitude Test.
The trouble with the SAT is that it isn’t fine grained or comprehensive enough for the top tier of universities. It is also not really good at distinguishing among majors.
Princeton could, for instance, have its own advanced math test for STEM applicants. That, and only that, would be a fair and relevant.
First, students should pass an emotional maturity test - what percent of time will the prosective student require a safe room, crayons and stuffed animals and what percent will they be in class vs whining about diversity.
Second, a life skills test - do they know how to study, do their laundry and make ramen noodles or is their only goal to party hearty.
If they pass those, then take the SAT.