Education is not "behind" it, as in causation. It's correlated. If you're "upper middle class", you can afford to send your kids to your/their pick of school, get them the right connections, raise them with financial and social sensibilities that will serve them well, distance yourself from negative influences, etc etc etc.
That special piece of paper is simply one of the common trinkets of that lifestyle. Stop assigning magic powers to it.
My parents technically earn below the federal poverty level. They definitely could not have afforded the expensive private education I received.
Yet from a young age it was always expected that I would attend college and a huge emphasis was put on education. Educational values endure across generations more effectively than wealth.
That special piece of paper is simply one of the common trinkets of that lifestyle. Stop assigning magic powers to it.