@angell84 You mentioned Germany having a problem with elitist private schools. I went to a state gymnasium school and my brother to a private school in Germany, so I know the German education system fairly well (or at least how it was when I left school nearly 20 years ago). My daughter is now going through the horrendous 11+ process in England, so I can compare the two education systems.
In my time, Germany had an absolutely superb state school education system. Depending on your grades in primary school (no horrible 11+ testing) you were allocated either a place at your local gymnasium school (for those with high grades), your local "real school" (it was actually called that - for those with slightly lower marks in primary) or "main school" (again - the literal translation - for those with the lowest grades). There were also separate specialist state schools for children with SEN. Depending on your performance at secondary school you had a chance to move up or down between schools if you wished (very rare to be moved down).
I can't remember any pressure or stress at primary about school allocations. You just went to your local secondary school. Although some gymnasia were older and considered to be slightly more prestigious- there really was no tangible difference between them. I actually first went to the oldest and most renowned gymnasium in our city (only because it was the local school) but was much happier when we moved areas and I joined a less famous local gymnasium.
Gymnasiums attracted the best teachers and standard of education received was very high. There were no behaviour issues or disruption to lessons because gymnasiums attracted the more academically able kids.
It was a great system and it was extremely rare (almost unheard of) to send your child to private school in Germany. My parents did it with my brother only because he really struggled at primary school and they were worried he wouldn't be able to cope in a normal school. Great decision, as he is now doing his Phd at Harvard. However, as I say, it was something almost unheard of unless you were an international student wanting to be taught in English.
Unfortunately nowadays the school education system has become a lot more problematic - largely due to people like the OP screaming that everyone should be equal and all should go to comprehensive schools (like in England). Well, now the kids who want to learn are in classes with those who throw chairs or don't speak a word of German. This in turn has forced parents to turn to private schools.