I don't know how things are structured in the UK, but in the US, at least, schools that are privately owned and run are not subject to any kinds or monitoring or testing or any kind of government intervention at all. And there are plenty of schools that are as cloistered as any home schooling situation and that teach the same kind of religious doctrine as some of the more extreme home schooling situations here. So in the US, just focusing on the home schooling situations would not work. The private schools would have to be subject to the same standards as the home schooling households.

I guarantee that ain't never gonna happen.