I did read your original and subsequent comments Noblegiraffe. I am sure you aren't the only one who feels that way, but it strikes me that no one opines that a shop worker who cares about the wider community should work in a charity shop, when the argument is no different. I am also sure there are plenty of teachers who are deeply committed to charitable activities that are entirely separate from their place of employment and others who do nothing charitable. It is never helpful to moralise about the priorities of other private individuals.
Moving on: Most non-state schools in this country are registered charities for historic reasons. Our understanding of what constitutes a charitable activity has narrowed and private schools have become increasingly expensive in relation to people's incomes, with the result that there is an important conversation to be had about their role in 21st century Britain. The perception that private schools are ivory towers is very damaging for them and they need to continue to challenge themselves on how to address it.
Having said that, I am not sure that reducing the financial burden of the Treasury should be a pre-requisite of charitable status. Bursaries in stratospherically expensive schools offer a clumsy, scattergun approach to reaching children whose families can't afford the fees. I also don't know whether it is a good idea for private schools to set up state schools or not, but I am pretty sure that coercion is not the way to go about getting them involved in state education. I am with the Chancellor of Oxford who has refused to get involved in the government's equivalent idea of making universities found schools. It would be a distraction from their core purpose and would probably involve them in activities that they lack expertise in. The same might be said (to a lesser extent) of private schools. I am not convinced they have expertise that the state sector lack, in fact I think it is more likely that the private sector lack the expertise that already exist in the state sector when it comes to transforming the education opportunities of children in areas that are currently poorly service.
As charities, private schools plough all their earnings back into the education and facilities that enable that education, so I am pretty sure that removing charitable status would increase their accountancy bills, but not increase the tax take for the treasury. There is no VAT on educational activities, which is completely as it should be. Can you imagine the nation-wide fury if parents had to pay an extra 20% for every ballet, football or music lesson that they take their children to?
So, I share some of your doubts about private schools in the state sector, but I don't have answers for what would be a better way of demonstrating charitable status and I don't think that removing their charitable status would offer any benefits to the wider community or treasury. I do think that keeping fees in check is probably more important than offering bursaries if private schools don't want to alienate their local communities (a 10 year freeze on fees and the facilities arms race might not go amiss!)
The OP is right to be sceptical about the particular private school's motives in setting up a school near her (given the timing and the geography she describes). However, it sounds like there is a need for more school places, so if local people are unhappy about the proposal they will need to provide an alternative solution - maybe a parent-founded free school that aims to continue providing the facilities that are currently offered on the proposed site.