This was my experience. In my SAT prep class 15 years ago, the first thing the instructor did was talk about the statistics of guessing and how even if you don’t know any answers at all, you can improve your score dramatically just by eliminating the obviously wrong answers before you guess. Then we went about learning ways to find the obviously wrong answers based on the format of the question that was asked. We also took practice tests, but it was the generic test taking rules that made the difference for me. I did very, very well on the SAT and then later, I applied the same rules to the LSAT and did much less prep and did very well on that too.

I appreciate your anecdote and that lines up with my experience. But a 10% increase is what I had in mind when I said I couldn't increase their scores much.

It doesn't seem to me to support the idea that the SAT tests mostly your (a) and (b). The average SAT at MIT is 1530 with the average at my local state school University of Indiana at 1250. Prep classes for the average UI student increasing their score 10% to 1370 wouldn't even get them to the 25th percentile MIT student.

It seems to me the SAT does capture something beyond simple ability to take a test.

If that were true then teaching how to take the test would have resulted in a larger jump on scores.