I wrote a long answer Martin and the internet failed me!
Even now, within the state system, schools run by local authorities have to declare their income, even the income generated by Parent/Teacher Associations.  There are things that such money can't be spent on - suppose the PTA of a LA run primary school wanted to fund extra teachers for instance. What could happen is that funding would be reduced by the LA because the budget was no longer needed. 
They can reduce it as much as they like - the parents of rich kids can more than compensate for that. Even if the funding was reduced to zero they would be able to fully fund the school - after all that is exactly what is happening now!
In the political debates, fairness can win, it just needs people to keep on saying it.
Not if the majority of politicians have a vested interest in it not winning - regardless of the will of the masses! (and I afraid I do not trust the politicians of any hue...)
If we take your attitude and just give up, saying that there is no stopping these people, that all outcomes where the poor get poorer and the rich get richer are inevitable, then that is what will happen./i]
Except that this is NOT my attitude... I entirely agree that we should work for greater equality - especially in the distribution of wealth. There is a book - The Spirit Level - which explains that, once the average income is above a certain point, the critical factor in all sorts of beneficial criteria is the ratio of the high(est) earners income to that of the median earners. Everything from health, happiness, crime, educational achievement and I forget how many other things they checked on all have the same shape of graph, with countries that have a large ratio having bad outcomes in all the criteria when compared to countries with a smaller ratio - whether that is caused by a smaller difference in the wage packets or by a large difference in wage packets coupled with an effective re-distributive taxation system. I just don't think trying to abolish the private schooling sector will achieve this...- and certainly not before persuading the majority of people who currently use the private schools that doing so is not the way forward.
If you want to get poor kids into the 'elite' schools the only way is for the govt to pay the full price for those places, but there are two problems in this.
The first is that there would have to be a LONG term commitment - at least 20+ years - and with increasing numbers of places funded.
The second is that, as I said earlier, poor kids would not want to be there as they would not be able to hold their own among their peers. Parents choose the schools they send their kids to quite carefully (at least, the majority do!); and there are a lot of things they look at - as you will know if you have kids. It is not only the academic standard of the school that is important. For a kid to thrive s/he need to be happy - and if s/he can not do the things his/her contemporaries are doing or s/he is 'the odd one out' for some other reason, then s/he will not thrive, no matter how good the staff are. Extracurricular activities (and the cost thereof - and even if there is a school savings system for these activities), any other costs the parents are expected to contribute towards, the ethos, how 'hard' the kids are (and whether they can get away with demonstrating this), the expectations of the teachers, the degree of parental involvement, are just a few of the many things we took into account when we chose the schools our three went to - and the net result is that they went through the state system, and all three are doing (or have completed) university courses - one with an MA under her belt, one currently on an M Phil and expecting to go on to Law School and then the Bar - whereas I went through the private sector (largely possible because Her Majesty paid for 2/3 of the cost since my father was an army officer!). On the other hand my sister's 2 boys went to the local private school, and did not do nearly as well academically - but I suspect that might have been the case whatever school they went to. I have nothing against the state sector, and I think there is a role for the private sector as well along-side it - but I do agree that it does need to have the pedestal it is on broken down!