To be fiar, tichy, having read through this zombie thread, I don't think the question was answered succintly, and to a large extend I don't think it was answered at all.
Firstly, it appears from the thread that some faith schools are funded differently than others. It appears that at least some church schools are funded partially (10%?) from the church.
From the Jewish schools I am familar with, the point was made that there is a "voluntary contribution" that funds the Jewish curriculum - so that the buildings and secular subjects are funded by the State, and the jewish subjects are funded out of the voluntary contribution. But a question that was asked repeatedly, and not, as far as I could see, answered, was what happens if parents can't afford to pay,. The simple answer is that they don't pay, and they cannot be discriminated against in terms of admissions or any other way. The schools do, in practice, have a reasonable idea as to who really can't pay, and who is merely claiming they can't afford to pay, and they do try to use whatever pressure they can, often social, to heavy those that they believe are freeloading. If you go on fancy holidays, or drive a fancy car, or own a fancy home (or even middle class versions of these), you will get a huge amount of social pressure if you claim that you are not able to pay the voluntary contributions. But some parents clearly don't care, they jet off on overseas holidays and they don't pay, and one of the jobs of the governors is to try and work out who it is reasonable to put pressure on, and who it isn't. On the other hand, the state faith school my daughter attended in Years 5 & 6 had unquestionably quite a sizeable group in her year that just couldn't pay. A group sizeable enough that they went to the head teacher and asked that the residential trip in Year 5 not happen, because they couldn't fund it as well as the Year 6 trip, and they didn't want their kids to miss out, so please would it not happen (and it didn't, and neither did the water research trip, day trip). And then when it came to the Year 6 trip, that nearly didn't happen, because there were enough parents who went to the head teacher and said they couldn't fund it as to jeopardize the numbers on the trip, until some of the other parents put their hands in their pockets and funded not only their own kid but the amounts that were missing so everybody could go.
So the short answer to the "what if parents can't afford to pay" is that the school either solicits charity from the wider community, or charity from the parents in the class/school, or makes the voluntary contribution for those who can pay large enough to cover those who can't. At least in the schools I am familiar with.