According to Washington Post and Slate, both being rather progressive, SAT prep might improve scores 10-20 points on average, with greater effect on the math section. There is a paper on the ACT website suggesting 30-60 points.
Downward adjustments for high performing demographics can be double that.
A cup of coffee would probably see similar or better improvements than test prep.
Khan Academy claims that doing their test prep is associated with a 115-point increase. Fortunately, their test prep is free, and is usable by anyone with access to a computer/phone/library.
I don't know about the SAT prep specifically, but much of their content is also downloadable for offline use, which is pretty cool.
Among admits, there are small differences in actual test scores especially among whites and Asians.[1] Also, there are racial gaps among actually using SAT prep.[2] Whites may actually be the least likely to use test prep courses depending on which source you look at.[3] I guess if you are applying to Harvard, test prep could actually be pretty significant.
When Harvard makes it decisions, test prep could actually be a major factor in admission there since scores are so close(assuming the numbers you gave). Especially since they try to reach out to people in different regions of the US.
The Slate article says the points effect is modest but also that a few points make a big difference to college admissions at the selective schools.
I also question how the controlling is done. The control group does better seemingly from doing the test more. You'd think that's one of the things that shouldn't be controlled for. After all isn't it part of test prep?
I don't know about the SAT prep specifically, but much of their content is also downloadable for offline use, which is pretty cool.