Unfortunately this is only true in anecdotes and is becoming less and less true as the US economy morphs into a much more unequal distribution of the vast wealth it creates. The Hamilton Project at the Brookings Institution has put out an excellent primer on the topic and here are the graphs that speak thousands of words.
This first is a alarming illustration of the increasing inequality of the US economy since 1975. In a wealthy and growing economy such as the US, should we not expect that the poorest children are doing better over time? This is a key metric of development in other parts of the world...
This next one rebuts the myth that it is all inherited genetics - smart rich parents have smart kids who get rich thanks to their own smarts and not the socio-economic position of their parents.
No, in fact, the amount invested the the future success of kids is rapidly increasing along with inequality. Especially as the public sector dis-invests in education, these enrichment activities are being privatized and this is likely contributing to the increasing income inequality and the decreasing mobility:
And these investments matter - the academic achievement of rich kids relative to poor kids is increasing rapidly.
To paraphrase Rawls: the true test of a just society is on into which you would be willing to be placed at random. Would you take your chances in the US?