Your personal experience is contradicted by some systematic studies. Unfortunately I don't have the citations nearby, but I'm sure if you search Google Scholar you'll find that you are an unusual person.
I think you're overestimating the extent that SAT prep courses play in college admissions compared to, say, personal tutors or involved parents who have knowledge about the particular rituals needed to get into good schools.
I should add that while it's true a child can visit the library to study for the SAT, it seems that's not very common.
I looked up some articles about SAT prep, reports say preparing for the test increases scores about 25 points on average. I'm not sure whether that was from the 2400 or 1600 test.
