Is there good evidence this makes a difference in the outcome of their children?

This is interesting but I'm not sure it supports that moving to affluent areas will help scholastic achievement.

For example, looking at the "median household income" feature, it doesn't seem to correlate well with "fraction of college graduates".

It might not need to, as long as median income does. The saying of “it’s not what you know, it’s who you know” and “you are the company you keep” rings very true in my experience. Just being around the kids of successful parents should help in opening more doors.