The good news is that test prep seems to have a very small effect: https://nepc.colorado.edu/sites/default/files/Briggs_Theeffe...
For some strange reason they are often summarized as "test prep had no effect".
But yes. If only because of the time constraints of the test (and it's more than that), sitting down and taking the SAT cold would be really hard for most people without prior exposure to similar tests. Heck, I'm pretty sure I couldn't today. I'm guessing I would do pretty well on verbal but too much of the math would be way too far in the past. And I wouldn't have time to derive formulae even if I remembered enough first principles to do so.
Test prep certainly helps. Knowing that test prep helps is key knowledge that is suppressed by these studies. Having the time to engage in test prep is often a luxury. Having the skill and discipline to do effective test prep is a lot to ask.
This is part of what colleges are selecting for.
Of course, heaving learned the material in the first place and doing a modicum of test prep so the test isn't a wholly unfamiliar thing is still important but it's probably fair to say that the SAT is far less gameable by spending money than many things.