I've been pondering over a few misconceptions on this thread.
Charitible independent schools (CIS) are charities with the aim of advancing education. Their aim is not to relieve poverty, so a child from a poor home is not more entitled to a place than a child from a more middle-class home.
Schools (CIS, that is) don't have a choice to opt out of their charitible status. For good or bad, they are stuck with it. The trustees do not have the power to sell the assets of the school, so they can close down, sell the land and buildings, to a new enterprise. What the Charities Act does is to order the schools to demonstrate their public benefit, and there is no one definition of what this might be.
Bursaries are an accepted way of demonstrating public benefit. They don't have to fund everyone who wants to join the school, but have to give everyone (within the aims of the school) the opportunity to benefit. This means that numbers of reduced cost places are available to everyone, but they have to compete for them.
Bursaries, if they exist, have to be available for a substantial proportion of the public - not just the very bottom of the economic scale, nor should they be so token as to cater for those who can manage if they cut back a bit (eg 5% off). When a family faces new financial hardship, they are just as entitled to the opportunity for a school place as a child that has never had money.
It is certainly not the child's fault that the family has fallen on hard times, and it is reasonable for a school to see what it can do to maintain stability for that child.
The aim of a school is not to relieve poverty - it's to provide education. Public benefit is derived by giving the opportunity for a means-tested bursary to all of those who cannot otherwise afford it. Most people cannot afford private education, and if you limited your bursaries to only the poorest, most people would still not be able to afford it, and you would not be able to show public benefit.