I can easily imagine that abject poverty would mess with SAT scores (or probably even prevent someone from taking the SAT), but for the majority of the country I would expect the SAT levels the playing field rather than hurting it. For example, I live in a rural town in Missouri. No one here is rich, but the majority of the town is not in poverty. I can't think of a better way than the SAT for a precocious child here to get into a top-tier school.
The value of the alumni and professional network of these lesser schools, as well as the brand name, is lower than the top tier schools, but the parents often did attend a more elite school, and their network offers inter-generational benefits.
It's interesting that state schools like Michigan accept a large number of less competitive in-state students, thereby providing the sort of springboard you mention, while also attracting a top tier of students that is on par with those attending Ivy League schools.