I've done a search to see if this has been done but couldn't find anything, apologies if this is previously-troden ground.
This is a really interesting report from the Sutton Trust.
Basically they are proposing the idea that Independent Schools should have the option of joining an Open Access system where there is a needs-blind admissions policy for any child who wins a place via competive entrance exam. If your child gets in and you are wealthy, you pay full fees. If you are poor, you get all or almost all fees paid. They have done a seven year trial in Liverpool with very interesting results.
The sutton trust reckon that per-capita on average it would be no more expensive than state schooling because of the proportion who would pay most or all of the fees, and would be less socially divisive than the current system as the cream would be skimmed off from the comp system based soley on academic grounds rather than academic social and wealth grounds as is currently the case. They also say that they would propose it was limited to no more than 1% of each year's intake such that it wouldn't have a significant effect on the ability mix of the non-selective schools.
I thought it was an interesting idea, but it's clearly a huge step from the current system. Could it work? Would people love it or hate it?