No? I don't follow your logic and I think it has to do with the intended goals. Is the goal to have some measure of "proportionality" of public versus P/P schools that win at each level of the tournament? If that is the case, I full stop reject that premise. And if one class isn't enough, how would a multiplier be a better solution than simply letting the success factor logically place any program, public or P/P, in the appropriate classification based on their performance on the field?
And to your post on free and reduced lunch: again, I understand the logic, but just as I find it wrong to value a student-athlete who attends a P/P more simply because they attend a P/P, I find it equally wrong to essentially devalue a student who is on free and reduced lunch. Again, that is not to deny the socio-economic factors at play with both populations, they are there, they are real, and they cannot be denied. My issue is how do you quantify those advantages and disadvantages in a logical meaningful way? Ultimately, I cannot conceive of an answer besides letting results on the field determine those advantages and disadvantages; hence the success factor.