it would solve themselves. all of the good PIs at the good private schools would go to the good public schools. there's no actual inherent reason why private schools are bad like you're implying.
Setting that aside, why do you think it is even desirable to limit funding to public institutions in the first place?
Seems like you're starting from this position and moving backwards.
If Public schools are superior, why haven't they already out competed private institutions? They have a pretty huge advantage from additional tax revenue.