It's a good question. SAT scores correlate with IQ tests as much as they correlate with subsequent administrations of the SAT.
Part of it is just that if you're going to administer a test multiple times a year to millions of people with a lot riding on their score, you need to constantly be developing new questions. IQ tests used by psychologists don't need that much churn, and the stakes are rarely anywhere near as high as on the SAT. Other part is that the SAT is specifically tailored toward the core curriculum of American secondary education, and is not officially supposed to just be a proxy for intelligence, even though it actually is. Colleges like to pretend that the SAT is testing "preparation" and "readiness" for their curriculum, because it's considered kind of crass in current times to admit that you're choosing winners and losers in part based on an innate quality that some people have a lot of and others severely lack.
