How are public schools compared to private? Does the quality of public schools vary a lot depending on which neighborhood they're located, like in the US, or are they mostly on the same level throughout a city/state?
Also, we have some national rankings of schools (both private and public) but don't get too hang up on that. My kid used to go to a very 'fancy', top ranked and expensive school before. Top of the ranks. Cost per month was over 700 euros. He was bullied there. Robbed of toys and things. His teacher couldn't get him to participate at all (he was identified as 'gifted' with an IQ of more than 130) or do anything in class. Talking with other parents, their kids had similar things happening (other kids stealing things). School tried to hide it until I threaten them with having the police involved (not exaggerating, I had 3 meetings with teacher and director person, and nothing was really done except vague promisses, on 4th time, I had meeting with my lawyer, I went to the meeting and said due to this and that, and the fact that what happening was crime A and B, and if it ever happened again, I would press charges both against the kid and the school (there is law where School is legally responsible if they are aware of this happening and do nothing). 10 minutes after leaving that meeting I had the school director/principal calling me to schedule an emergency meeting. After that meeting, it took me 2 weeks to move my son to another school.
He is now in a 'shittier' school, even has more 'problematic' kids, and there isn't a single issue, he loves it. He may not have fancy digital dry boards, and music lessons with some famous musician, but he is much happier kid. Just did his 'exams' and except for Portuguese (he hates it) scored almost max points in everything. Just because a school charges a lot and has all the 'fancy' things, doesn't mean it is the best for the kids
edit: P.S. I agree with you on expensive schools not being automatically better. A friend of mine had a similar experience to yours, and it was a very expensive school.