I can only assume that people who thing that private school children don't have links to their community must assume that all children live in tiny bubbles surrounding them and their school???
DD goes to an independent school 3 miles away from home. She has no school friends within walking distance but has many play dates, etc.
She goes to brownies at out local church hall and mixes with other girls of the same age from our local area. She plays with some of these girls outside of Brownies, as a reault of meeting there, now.
She goes to Drama about 4 miles or so from home, mixing with boys and girls from other local areas. So even more of an increased friendship group in anyother community.
She goes climbing with boys and girls in yet another area - making even more links and friends.
She plays outside on the street with all the other children - all live on our estate, but go to various schools including the catchment school, schools out of catchement and the other independent schools. Where they go to school has no bearing on who they play with at all round here. They all muck in and go out and play.
She has other friends from other activities or links, all over - not just our local community. Her best friend lives about 3 miles away, not related to school at all.
DD doesn't live in some little school based bubble. She is really sociable and firendly and will just make friends with anyone and everyone!