The manner in which school fees increases have outpaced inflation in recent years is one of the issues discussed on the "level of family income required" thread. It was also the subject of a leader 10 days ago in The Times, which accused schools of abandoning the middle classes.
The consensus on the other thread was that the "nouveau poor" (salaried doctors, dentists, engineers, accountants) didn't deserve to send their DC to private schools any more and we must just face up to the reality that the market can bear their fee increases, i.e. a market which can pay them out of bonuses, profit shares, "little inheritances", and the generosity of super rich grandparents.
I feel sorry for parents who calculate they can afford fees and then get hit by rises like this, which bear no relation to their own rises in income, if any. I also feel sorry for the schools, which are squeezing out the middle and becoming weird polarised communities with a smattering of bursaries and a majority of pupils whose fees are paid from sources, which couldn't be regarded as "earnings".
Pension contributions and other staff costs have risen but schools could choose to mitigate those by increasing class sizes a tiny bit and foregoing other luxuries. They choose not to because they can get away with it and because they don't care.
There are some honourable exceptions, even in London.
Oxford, unfortunately, has been afflicted by this pandering to the super rich ever since Andrew Halls was head of MCS. He's now at KCS Wimbledon where he was heard to say on a BBC documentary that Tatler's schools guide was "important". Those are the values of "leading" heads these days. We can choose to reject them.