And to emphasise the difficulty. It's estimated that assortive mating (ie smart professionals marrying smart professionals) increases inequality by around 25%. Those couples tend to be more stable, with something like 9% of university educated women giving birth out of wedlock per year vs 61% of high school only women. When they have children, professional couples tend to expose them to around 32 million more words by the age of 4 than children of parents on welfare.
That's just up until they're 4, and it's not due to money or opportunities (which no doubt come later on when wealthier folks can move to areas with good schools or whatever), but good parenting/life skills.
That isn't saying poorer kids are at fault or they deserve anything, it's accepting that parenting plays an enormous role in the wellbeing of a child, and their future success as an adult (as the rewards for being highly educated are bountiful and the circle tends to perpetuate of marrying other educated people and so on).
It might not be impossible, as you say, but I'm not sure it's at all easy either.