'The only sad thing is how few people know about this. Under-privileged students think they can only afford state schools.'
This is a huge problem that, in retrospect, was rampant where I came from. A pretty typical middle-class suburban high school, but one which rarely to essentially never sent any graduates to Ivy League schools. Why? Because it wasn't part of the community's culture.
How do you fix this? In anything other than upper to upper-middle class suburbs, the parents themselves probably went to state schools and have middle class jobs (which themselves are under pressure now). So the parents and neighbors didn't know anything about Ivy League schools or how to propel their own kids into the upper class.
By the time most kids learn how the culture works, it's too late, they're already in whatever college and life track they happened to fall into. It's takes especially thoughtful parents or other older role models (like great teachers or counselors) to see a kids potential and tell them about their life options.
Some kids have this experience (maybe it's a family friend or distant relative that happens to be very wealthy and shines a light on possibilities to the kid), but most do not.