I taught/tutored the SAT pretty consistently about 8-10 years ago. In general I would promise approximately 300 (!) points of improvement on a 2400 point test, and I would generally deliver on that promise, assuming they weren't starting from a place above 2000.
Of course, I was charging anywhere from $50 to $150/hr, depending on the family. So people were directly trading dollars for points. You can try all you like, but there's no method I know of to make a standardized test that doesn't benefit from a knowledgeable tutor giving you a big leg up.
It's not like eliminating the SAT really changes this, though. Students from higher socioeconomic strata are going to have better environments and access to learning. The hope is to invest a huge amount of money toward after-school programs, teachers' salaries, and childcare assistance, but America is pretty garbo so that's probably not going to happen.