echt: you can't just "remove" charitable status. If you could, many schools would simply have done sows the financial benefits are not that great.
But it must be maintained as it is the charitable trust that owns all the school assets. They can only use those assets for the charitable purposes for which it was set up. So if the school wanted to be a private business, it would have to move - with zero assets, and the current premises and assets redirected by the trustees to similar aims to the founding intention.
Up until the Labour administration, all charities were obliged to follow the charitable purposes in their founding arrangements. Labour introduced new conditions, unrelated to the existing charitable aims, concerning poverty - a condition they do not appear to have attached uniformly to all charities. This was the result of neither change in the law, nor legal review. The rather nebulous status of the Charity Ommission's new stance is what has led to the possibility of legal challenge (don't think anything legally vonclusive has happened yet, but there is a prima facie case that the Commission has acted beyond its mandate).
Personally I think it was a very cynical attempt to reintroduce the Assisted Places Scheme, but with private school parents picking up the tab rather than the taxpayer. This is failing as a) a good proportion of parents are not that affluent (esp Forces families who would be disproportionately hit) and b) despite an expansion of both provision of bursaries and advertising their existence, there don't seem to be that many new potential pupils who might be able to use the opportunity.
Cognita etc love this possibility as it gives them a chance to buy school buildings (possibly on the cheap), though of course depending on local planning authorities they may be permitted to change use.