I'm not sure I'd say I'd like my children to go private as I think the state schools round here are pretty good and DD1 is doing well at her secondary. I would say that I think DD2's interests and talents might be given more airtime at a private school but that's unlikely to be an option when it comes to secondary school.
We've always had a nanny, nannyshare or after school babysitter instead of using childminders or after school club. This has meant that music practice can be supported at home, extra curricular activities of their choosing can happen because the childcarer is employed by us, in our home and so can take them there/back. This makes a massive difference when it comes to facilitating participation in a range of activities.
In addition, we have encouraged the DDs to go on residentials that meet their interests whether that's dance, something instrumental or another hobby. At times, this has involved negotiating time off school for them as many of the courses are set to coincide with the private school holidays (and many use their facilities). These don't tend to come cheap but they're significantly cheaper than 7 years of private school!
DD2's main hobby seems to be dominated by privately educated children and the fact that she's not is a non-issue. I would say one of her strengths is the ability to hone in on what she has in common with another person and to use that as a starting point. I've no doubt that she'd benefit from opportunities at a private school but because she's good at the musical instruments she plays and the sport she takes part in, is unfazed by being in an unfamiliar setting with complete strangers, is genuinely interested in others and willing to have a go, she seems to be offered extra opportunities and experiences.
DD1 is much more socially intimidated and her main hobby is dominated by boys in private schools. I'm not sure the self-confidence she lacks in those scenarios would be improved by being at a private school so we just continue to encourage her to take part and to hear the compliments she's paid instead of assuming they're out of sympathy
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