I agree with you on private schools. My other half went to a private school, and her parents were in no way wealthy. She left state education because her primary school messed her education up so much she (alongside other kids in her class) failed her Year 5 SATS and refused to take the blame for not teaching the class adequately. She went from failing in Year 5 to getting a first at uni - probably thanks to a solid private education.

With that being said, the current policy doesn't seem to be to scrap them entirely, but to remove their charity status. This does make sense, but will require more thought than what they've given so far. Removing their status ensures they're paying fair tax, but it also means that they are no longer required to share their facilities with state schools.

It's a weird policy to lead with, and the cynic in me believes they're doing so to push the class divide, in order to get more Leave voters behind them.