"the problem I have here with all this discussion is that there seems to be an assumption that it is a reasonable response to unfairness to tell people that they are not allowed to provide something for their own children, if not everybody else can also provide it."
This is, of course, an important point. The freedom to help your children as best you can is important, and not to be trodden on lightly.
I am not suggesting that whenever a freedom produces any unfairness or inequality, we should take that freedom away.
For example, I am not here arguing that people should not be free to, e.g. teach their own kids to read, on the grounds that this causes an unfairness and inequality - i.e. because other parents are unwilling or unable.
But "freedom" doesn't trump all other principles on every occasion.
Surely, where there is a very great unfairness caused, and indeed, serious harm being done, by people taking advantage of a freedom, there can be a good case for curtailing that freedom.
For example, we don't allow the richest parents to buy their children all the best university places (hmm, well, having taught at Oxford, I'm aware that, on occasion, we do - but of course the practice remains frowned upon and rightly so). This is a "freedom" we don't permit. And for very good and obvious reasons.
But notice that it is exactly the same sort of reasons that motivate my suggestion that we take from parents the freedom to buy their children all the best school places.
"Freedom" doesn't automatically trump the kinds of considerations I am raising.