"Evidence shows that privileged families will stop at nothing to prevent their children being overtaken [by anyone else's children]."

Society needs social mobility so that we don't end up with incompetent aristos in charge and so that people from poor families get heard. But social mobility is not a solution to the problem of poverty. The small chance of becoming wealthy is of no benefit to the poor children who don't become wealthy (which is most of them).

Growing up in poverty and hardship stunts children's growth, including their mental growth, so when judged on their "merits" later they will be consigned to poverty.

The solution to poverty is to redistribute the wealth and income. When no child grows up in hardship, no child will be stunted. Indirectly, this will help social mobility too.