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.
I'm skeptical of the idea that money and fix any problem, and that there are no inherent differences.
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.
When building a highway, the government should put out notice and buy the best offering.
Im not sure why you would want government buying sub-par research just because it comes from a public school