This I very strongly doubt:
This case involves people forking over fair amounts of money to get into schools with greater prestige(amounts such that the price difference between private and state schools, not that it’s always all that big if you are looking at out-of-state tuition, isn’t an object for them; plus they are mostly trying to cheat into the more expensive ones).
It’s already the case that state schools vary widely in prestige, both between States and within individual states’ systems. How well correlated the prestige differences are with quality differences I don’t know.
If we eliminated the private schools how would that change this dynamic? They’d be trying to cheat into the high prestige state schools rather than high prestige private ones(some may have been already, I didn’t see a full list of target schools and there are some very well respected state schools).
If you want to talk about how underfunding of state schools(and some perverse spending priorities on important things like administration and sportsball arenas) makes it unacceptably difficult to get a decent college degree for less than a small fortune I’m all ears; but claiming that people cheating for prestige can be solved by picking off some of the current high prestige schools for being private is just wrong; especially clearly so when various state schools already have well known and commonly recognized prestige levels attached to them.
Having Mommy and daddy buying mediocre rich kids into UC Berkely vs. Stanford isn’t much of a difference.