Very few if any middle classes can afford private now. By affluent you must mean stinking rich.
I don't think is a very helpful characterisation of those who attend private schools: about 7% of children, rising to a higher percentage of sixth formers.
If you have one child and low accommodation costs, private school can still be affordable on relatively modest incomes e.g. 2 x teacher salaries. One of my DC's friends is in this situation: parents had one child in their late 30s, had paid off mortgage on small 2 bed house by the time she started secondary. Most people would not call an income of 2 x teacher salaries "stinking rich".
I have a student who attended a private school for sixth form because the local school did not offer further maths. He had a bursary and his parents remortgaged their 100k house to pay their part of the fees. Again not "stinking rich".
I would think that the majority of parents of private school children are in the top 20% of incomes nationally, but there are a significant number of children from backgrounds such as my two examples above: modest incomes, family makes sacrifices to pay fees because of issues with local schools. There are also many families who have good incomes but are not "stinking rich" and who don't take any holidays, drive old cars etc to pay the fees.