OP - what do you think the biggest influence over behaviour is? Behaviour is usually better in private schools - does that come from the home lives/parents or from the school having stricter rules or higher behaviour standards?
This is a tough one, as you do get shocking behaviour in independent schools. Children of arrogant parents can be arrogant, and they get back up from home. You also get poor behaviour from children with SEND because you are not meeting their needs at that time.
Inherently good behaviour comes from children whose families clearly value education and recognise that achievement means hard work, and are willing to do their bit at home (eg by eating with their children and having conversations about current affairs).
I think having clear family values builds character and resilience, which helps with learning.