A poor child of high intelligence, born in the U.S. (or indeed almost anywhere in the top 80% of the world) has greater access to knowledge and the tools of self-improvement than ever before.
Nowhere does this article mention difference in ability or intelligence. Search the article -- the words "ability" and "intelligence" do not appear. You cannot address the subject of income or social mobility in an intellectually honest way without engaging this subject or controlling for or studying these variables either on an individual level or across groups or populations you wish to study.
Of course, if these results are just the natural result of genetically hereditary features like intelligence than the conclusions are insanely gloomy- due to the "assortative mating" trend cited in the article, the US seems to be doomed to have hereditary social stratification.
We have had hereditary social stratification for basically all of human history. Only in recent centuries and only in the most developed parts of the world have we taken a brief detour away from some form of feudalism which is observed pretty much everywhere people settled down to practice agriculture and produce surpluses. It seems like today, we are busy recreating Feudalism 2.0 with American characteristics.
The wealth inequality we observe today seems fairly indistinguishable from that which we might observe in a world where genetically heritable features (not necessarily or only IQ) largely determine financial outcomes and where wealth is inherited, compounding effects through generations. Whether nature or nurture, advantages will be passed along even if we implement a 100% inheritance tax.