It is your child- and it's also your family.
The difference between private school and state school+extras is basically that with private school you buy a package: enrichment activities, sports, leisure activities are all provided on the premises. This suits some families but doesn't suit others.
It's like an inclusive holiday: if you are the kind of person who wants to hand over a cheque and know that everything is being taken care of, then that would seem to be the optimal solution. ]
Other people prefer to make their own travel arrangements and if you have that mindset, then a state school + extras can work very well.
I am pleased that all our education money is not going on school fees because it means we still have the money to pay for enrichment activities that even a private school wouldn't provide, to maintain a very good family library, to do trips that are specially geared to our interests and needs.
Basically, state education to us partly means retaining some control over the educational spending: a private school (assuming we could afford it) would mean handing over all the spending decisions to somebody else.
It means if dc show signs of a special talent we can focus the spending there. Like my parents were able to send my db to take lessons from the Kapellmeister of a well known symphonic orchestra: coaching on that level wouldn't be provided even at private school. Or I can let dd audition for the best youth theatre programme in the country because I know if she can get in I can pay the fees.
I just find it a bit odd when people say you have to go private because a good education is more important than anything else. My children are getting a good education. It's just not all happening at school and being taken care of by other people; it's a combined effort between the school and us.