I think the mistake people make is thinking people move to private schools because of swimming pools and concert halls. No they don't. In the main it's because (a) the parents were privately educated and want that for their kids (b) they perceive that the education is better or (c) they have had a disappointing experience at their previous school (often state).
No one I know talks about the fluff around the edges as mattering. Yes it's convenient not to have to drive my kids to many extra curricular sports because they can do them at school but I still have to anyway and it's not a factor in why we go there, or anyone else I know.
It adds to the amenity of the school. Most state schools in our area barely have room for the classrooms let alone other facilities.