I was one of those kids. I never had a stable home, I have 5 siblings with many parents, I moved between different parents, lived on the floor with mom and siblings at one of her friends homes for a few months etc. But mustering up the energy to put in effort for one test is still possible, doing that for everything, no way.
Kids aren't stupid, they have spent years doing things, reading things, it isn't hard to find out information about college, SAT etc. A bad home doesn't mean you are stupid even though you make it sound like it does.
I also know people who've been in your situation, and that's what I see more often than not.
I can understand if you think that's what the SATs are for though, a kind of lifeline for kids in bad situations. I just have reservations about it actually working enough of the time to not simply be another opportunity hoarding play by wealthier people.
The main effect of removing it is that disenfranchised smart kids don't get into college, replaced by dumb kids with stable homes who spent a ton of time studying yet were too dumb to score well on SAT. Colleges are already full of such kids, and so is the job market, we don't need more of them.
People who study a lot study a lot for the SAT, people who study little study little for the SAT, and in between people do an in between amount of SAT. That somehow seems like a good starting point, which we'll probably be stuck on given evidence will be hard to come by.
FWIW there were kids in my kid's class who had three tutors and studied every day over the summer holiday (the test is right after). They would have stopped if it was so obvious the benefits stop past some level. Just as there will be people who think it's a great idea to put 1000 hours into it, there's going to be people who think they'll just show up on the day and see how it goes.
Edit: seems this wasn't a direct descendent of the the thread I thought it was, about the UK 11+, but the point is the same.