The logic is airtight but utterly inconsistent with all evidence. EPI summarizes a large OECD meta analysis of social mobility studies, finding that the United States has poor social mobility (lower number means more social mobility):
Matt Bruenig adds some context that makes plain just how terrible social mobility is in the United States:
So, you are 2.5x more likely to be a rich adult if you were born rich and never bothered to go to college than if you were born poor and, against all odds, went to college and graduated. The disparity in the outcomes of rich and poor kids persists, not only when you control for college attainment, but even when you compare non-degreed rich kids to degreed poor kids!