I don't think this is just because the SATs are bad tests but fundamentally emerges from the institutional single source of truth model, where the incentive is for the institution to make their test as general/broad as possible and the most broadly accepted , as to capture the most students and hence fees.
I believe this would take the form of a dedicated assessment tool that allows teachers to create individualized assessments JIT based on - or in concert with - their planning and performance workflow. If I'm teaching a structured lesson on polynomials, I should be able to create a valid assessment simply be requesting that one is made. My students should be able to take that assessment on the spot and the results should be immediately available.
I will take assessments with 80% confidence of validity 10x/week over assessments with 100% validity once per year every. single. time.
There's more to be said here - but that's the gist.
College Board owns SAT, AP, and (essentially) Common Core so there's been a trend to tighter coupling. I like the idea of an AP Test that anyone can and be assessed separate from the teacher. Same for SAT. I think we need more of that model, not less.