I don't think you can ever eliminate that, short of removing kids from their parents Plato's Republic style.
Adding more rules will just create perverse incentives (e.g. parents incentivized to increase crime rate in their neighborhoods to boost their children's scores.)
I agree that you can’t equality of opportunity without the government taking over the task of raising children. But that undermines OP’s premise. If “equality of opportunity” is not an option, then “equality of outcome” should be on the table.
