"Clients have little say over how finances are structured."
Parents, parent governors, governors appointed by a church or university, former pupils, prospective parents, the Charity Commission, all of these stakeholders can influence how a school structures its finances.
I urge MNers to scrutinise school accounts on the Charity Commission website precisely because they can reveal how a school does structure its finances and what it actually spends its money on. Accounts can expose a school's hypocrisy about bursaries and about its alleged attempts to become "needs blind".
There might be an appetite for the kind of scheme you describe, maryso, but I hope no London day school (to get back to OP) ever tries it because it could easily be destroyed by it. All it would take is a political, economic or fiscal change or an enquiry by the Charity Commission for the whole delicate structure to collapse, reliant as it is on the uninterrupted confidence and commitment of a small number of super rich parents.
If anybody who is unable to afford fees of £60k per child per year is eligible for a bursary, you will be hard pressed to find enough parents who can afford that much to fund all those bursaries while their own DC go through the school.
When schools appeal to former pupils for funds, they like to aim for those who don't have children themselves and those whose children have flown the nest: they are the most likely to have disposable income to spare.
Anyway, "needs blind" admissions at Harvard have not made it as accessible as it pretends: the average income of parents of students at that university has been calculated to be $450k pa, which is the average income of the richest 2% of American households (Piketty p778).
I'm in favour of independent schools fulfilling the duties imposed by their statutes, offering free places and building up bursary funds but keeping costs and fees at moderate levels, in keeping with principles on which they have operated for generations, maintaining the loyalty of stakeholders and earning respect.
Paying a head £300k pa (NLCS) and raising fees by 37% in 5 years (SPGS) are grotesque. These aren't "market rates". They're the products of greed and stupidity. Frances Mary Buss and Dean Colet have been betrayed.