I think the question doesn't quite work for most areas, because the saving on school, fees wouldn't be enough for a mortgage free house - so the choice isn't as stark as suggested.
Personally, I always say, don't pay for Independnet unless;
A. The individual independent available to you offers significantly more than the state available to you. (So, if the state is very very poor and the independent is very very good then it is more worth it, but in many cases, the Independnet isn't actually significantly better nor the state very poor, so the clear cut gain is small)
B. Most importantly I would say don't pay unless you can do it without it making a major impact on your lifestyle. Mor most people it's not worth going without holidays or money for a new boiler etc to pay for this.
Personally, I would want to have a good chunk of money in the bank if paying fees - partly as a back up for fees, but also for all kinds of emergencies and the future. I wouldn't consider fees unless I had this.....and actually this would probably then allow me to gift a deposit contribution to my child....so I don't see it as an either /or choice, but that those paying fees often are able to give deposit money too.
I wouldn't be interested in a mortgage free house gift. I don't think it's beneficial to not have to make any payment for your accommodation. However, I would be interested in helping a child get together their deposit so they can then lay their mortgage themselves.
And if I could pay fees without it making a big impact on my lifestyle and if there was a great Independnet locally, then I would pay. I don't just see the value of these particularly great schools (not run of the mill 2nd raters) in terms of the exam results or the jobs people later do - so purely in grades or finances - but more broadly in terms of less tangible benefits such as opportunities to foster interest in a broader range of areas or depth in a certain area - the experience and journey itself, which can make a less obvious impact on later life). Being able to choose for these things and not just grades or future job is indeed a luxury and I think why many people for whom money is no object do choose Iindepenendt. It's not always easy for those who have no real experience of this handful of very top schools and who only measure in terms of grades and salary paid by the job later chosen, to see what the benefits might be, especially compared to a good comp or state grammar.