Public benefit must be contained in the purposes
Where the purposes of an organisation are ambiguous or unclear, the court may be able to determine its true purposes by reference to its activities
But there is no legal basis for the notion that an institution whose purposes are charitable (and so for the public benefit) can lose such charitable status merely because of the manner in which it is run by the directors or trustees
Under the law (both before and after the Charities Act 2006) public benefit is contained in an organisation?s purposes, not in its activities
The purposes of a charity for the advancement of education are necessarily for the public benefit
This is a cut and paste from a presentation made by Peter Luxton who is a charity law specialist and professor at Cardiff Law School, Cardiff University
"A charity does not have to show that it is providing public benefit in the way in which it carries out its activities
A charity merely has to show that it is carrying out its purposes
A charity asked to state how it has been satisfying public benefit in carrying out its purposes could therefore respond that it has been satisfying public benefit in that its purposes are charitable for the public benefit, and that it has been carrying out its purposes in the following ways (and so listing its activities)"
I know it doesn't really answer the OP's question. Because the law is often an ass. But this is a legal specialist's intepretation of the law as it stands. I doubt we will ever see a school stripped of its charity status so that we can have a proper test case. It's all New Labour puff.
ANd there are some ridiculous charities. I wouldn't want to give my money to a lot of them but I don't resent them receiving money from the state just because they are minority interest charities.