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.
Nothing wrong with individual universities having their own tests for their own purposes; perhaps they could use the SAT as an “entrance to the Entrance Exam” test.
The GRE has subject area tests and a general test. SAT could do likewise.
Besides the SAT and ACT I took The Architectural Aptitude Test.