This in my opinion is the main problem. Take this "marketing tool" away from the private schools, and the rest of the problems are fixable (e.g. with diversified pupil compensation). But when you let a private business both grade themselves and sell entry to (free!) university, you will quickly have massive problems...
Up until the 70-ties or so there was a separate government agency responsible for final exams and grades. If we want to keep private schools (and I think we should at least try), then we need to reinstate that separation of education and evaluation, urgently. Problem is we now have a large swath of middle class parents who feel that they've been very smart and ensured a great future for their children by choosing the right school for them (and at least subconsciously they understand that the main reason is the inflated grades). They don't want that privilege taken away. There is no limit to the hypocrisy many people can accept if they feel it benefits their kids.
Source: I'm Swedish, but I don't have kids so not super knowledgeable. Just my impression after following the debate and talking to friends with kids.