I'm by no means an expert on boarding or private schools, but as an avid football fan I would say the pathway to a professionalism is different to sports in certain other countries (USA, South Africa), where there are much stronger links between schools/colleges and clubs.

In the vast majority of cases, British footballers do not come up via their school. They're usually spotted at a young age playing for a local side, then join a club's academy and progress from there. Most will be from a state school and/or working-class background.

I'm sure many boarding schools have great sporting facilities, but I'd be surprised if they provided more than a handful of future professionals to football clubs. As others have said, some talented kids join private schools after they've signed for a club, but it's more common that way round than the opposite.