The issue raised by OP is how those on middling incomes are to cope. They won't usually be able to get "help" from the schools: not only are their bursary funds too small to extend to parents who are doctors, teachers, university professors, and other professionals, they have also hiked the fees up at rates which bear no relation to salary rises (if any).
Twenty-five years ago when I was at school there were virtually no children of teachers or academics at private schools. There were plenty of doctors' children and plenty of lawyers' children - demographics were heavily skewed. Self-employed parents could fudge their incomes and use assisted place schemes but those who did PAYE with employers couldn't. Many "professionals" earned above the assisted place limit but were not in a position to afford fees.
At my DC's schools now there are still plenty of doctors' and lawyers' children. The most noticeable difference now relative to 25-30 years ago is that typically both parents are working i.e. fees are paid out of 1.5-2 professional incomes.
As I said upthread, my partner and I paid school fees on two professor salaries with a 2k per month mortgage, with plenty of money to spare. (Two times professor salary = 140+k). Even if one of us worked part-time we could have still paid the fees, on 1.5 incomes. In my DC's school it is simply untrue to say that most parents are super wealthy or get help from grandparents or have paid off their mortgages.
I really don't understand why private schools should effectively increase their fees to subsidise parents who are e.g. doctors via bursaries. Even assuming one full-time salary from a doctor, with a stay at home second parent, such an income is high by national standards. (It would also be somewhat dubious to give bursaries to families in which one parent chooses to stay at home when they could earn, at the expense of families such as mine in which both parents work full-time.)
All of the arguments by Granolamuncher seem very London based. Even if the demographics of SPGS have changed substantially, the demographics of my DC's schools are really not very different from when I went to school.
I do agree that academic scholarships are less common, though. My family was above the assisted place limit by some way but not really in a position to afford school fees for all of us. My siblings and I got substantial scholarships (50-100%). Nowadays very few schools offer scholarships above 20%.
By the time my youngest DC reaches secondary the fees may well be above what we are willing to pay. But so be it - the DC can go to a state secondary instead. Pricing people out of private education will as a knock on effect probably improve the local state schools.