A lot of these "good state-funded schools" pushed people to specific A-levels that looked good to universities, and had coaching on the university interviews and entrance exams (not to mention a much higher standard of teaching). These schools regularly had 100+ students a year get into $GOOD_UNIVERSITY. In contrast to my school that 2 people go in my year and none some years, and and couldn't even work out which entrance exams I needed to be entered for (which they had to do on my behalf).

I don't agree that the SAT is helpful though. These kind of standardised tests measure ability to prep far more than they measure intelligence. In my experience the universities are actually putting quite a lot of effort into making their courses widely accessible (setting lower entrance grades for people from less good schools for example). The problem is really with vastly uneven quality of teaching at the high school level which means that some students really are a lot more prepared for the courses than others (even if that doesn't reflect base level intelligence).