Social mobility when achieved through government policy rather than economic reality will cost more in the long run.
I think both theoretically and evidentally, this is not the case. If you are brought up with wealth, you have better nutrition, better education, more lucrative social networks, more useful free time, and a far less severe exposure to risk. If there is no government policy to redress this imbalance, then probability dictates that wealth concentrates and poverty, on the whole, becomes entrenched. And this is what we see in the modern US.
Individuals certainly have opportunities to make for themselves a better life. But if they are coming from a poor background, those opportunities are far fewer, they must work harder to take advantage of them, and the consequences of failure are much more severe. Essentially, the dice are loaded.
Moral considerations aside, a society where 80% of the population have the opportunity to take risks and be innovative and exploit usefully the extant infrastructure is going to be economically more successful than one in which only 20% do.