No, you cannot. IQ is highly correlated with wealth and is also highly heritable. So in all likelihood what you've observed is that smart kids win academic competitions, and also happen to be from rich families. If you take a bunch of kids with an IQ of 100 (or 115, or even 130) and put them into private schools they'll do a little better but they won't be winning any academic competitions.
That article doesn't seem relevant. 'Poverty impedes cognitive function' is entirely compatible with 'IQ is highly correlated with wealth and is also highly heritable', and with the claim that private school can't make a genius out of a kid who lacks an extremely high IQ.
edit: maybe I misunderstood, and you only were only pointing out that some potential-genius kids have that potential stifled by poverty. But that doesn't seem to be the original point of disagreement.
