I think all children should receive the best education possible, this means properly funding state schools and having good support for children who are struggling, have no family support, SEN etc., this is one of the main requirements to be able to create a fair society with equal opportunities (of course, it is not sufficient to this end)
If private schools did give an advantage to kids, I think they should be banned, they would indeed represent a way of buying privilege, and yes, therefore a form of cheating (I also think you can't blame parents for wanting the best for their kids given the options, though, so as long as we don't have universally good state schools I can understand those who can afford it sending their kids to private, that's why I think they should be banned)
As it is, the evidence is mounting that they really do not give you a better education at all. It is simply the wealth of the parents that is linked to better outcomes, not the school per se, so, no problem if you want to waste your money on a private education!