If SATs were some sort of general intelligence assessment, it is unlikely that a $500 Kaplan course would significantly increase your scores (which they do).
Like any other study, you need to look at the source of that data. I'd trust a study re: SATs sponsored by "National Association for College Admission Counseling" about as much as a study about pollution sponsored by the "American Petroleum Institute".
It's a small anecdote, but my high school got a grant to do a pilot program to incorporate SAT test prep into the school program back in the 90s. IIRC, the average score went up 100 points vs. the PSAT. With the old version of the test, I went from the 1200s (80th percentile) to the 1400s (95th). Writing was an optional test then, and the test prep didn't cover it, but I was already familiar with the writing process from AP courses.
30 points IMO would represent prep focused on test strategy exclusively. For example, with tests like the SAT, answering questions wrong comes with a higher penalty than not answering.
If you drill on vocabulary, tune your writing to line up with the scoring methodology and are familiar with the structure of math problems asked, you're golden. But knowing those things doesn't grant you greater general intelligence.
I don't think NACAC is equivalent to the API. It's not like college counselors are the ETS—in fact, many counselors complain about how much some students focus on the SAT.
As for your anecdotal evidence, it sounds pretty flawed to compare results on the PSAT to the SAT directly—I also got a much higher score on the SAT without doing any studying at all, probably because they're scored and weighted entirely differently. Moreover, if this comparison was done over a year (ex. sophomore to junior year), the results are likely even more flawed—there's too much confounding development in that year to attribute the increase to SAT prep.
Anecdotally, I know I did much better than all of my friends who spent months studying for the SAT and drilling on vocabulary, math, and strategy. If the SAT is so game-able, they should have overcome me.
