I have known state educated children who loved their school experience and state educated children who hated theirs. I have known privately educated children who loved their school experience and privately educated children who hated theirs.
It is a combination of so many factors: the school itself (as an individual institution rather than as part of a certain sector), the academic ability of the child, the personality of the child, the personality of whoever else happens to be in the class at that time.
Some things you have to accept you cannot foresee. When dd was 4, I thought she was a "normal" child: 4 years later she was a disabled child with strong pastoral needs. These days I would never consider a school that did not have a good reputation for supporting its weakest members, because I know how quickly the "normal" child can turn into the child with special needs.
Dd is state educated. Were it not for her poor health, there would be no question of her having to "do it for herself": her school offers a good level of education, with plenty of opportunities of enrichment, and there are enough other pupils with varied interests and ambitions to make her feel this is quite normal.
All the schools she has attended have had robust anti-bullying schemes, and there has been very little picking on clever children. It is a little known fact that any working class and lower middle class families do actually admire hard work and good manners: they want their children to get on and encourage them to believe that this is a good thing.
I'd say visit the schools and speak to people other than the head. A SENCO or pastoral support officer is often a good person to talk to to find out if things are really like in the glossy brochure.