It's much more complicated than that. For example in my experience a school per se doesn't matter much for many kids from problematic homes. What matters is that they get a time away from home. COVID lockdowns killed exactly that – they didn't have to be away from home and it was very bad for their mental health.

Then there is a growing number of parents who are actively working against the school, undermine the child's trust against school and teachers etc. Most of the time a school can't do anything with these and outcome is always bad.

Then there is a growing number of children who don't do well in school no matter what. They can have a good home and good school, but because of some neurodiversity for example do very bad anyway.

I also don't agree that good home alone is enough for good outcome. It really depends from many things.

In short – education, outcome from school and home etc are hyperindividual and there is no hard rules.