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It sounds like those "Steinway schools" are only a small number of schools. I think most of the day schools are keenly aware of how their fees compare to other local schools since they're drawing on a limited pool of nearby families.
Not sure where you got the £15k to £60k inflation either, the relevant quite seems to be this one "independent school fees have risen by 55 per cent on average over a 20-year period since 2003 to about £15,200 per year. In some of the larger schools, such as Eton and Winchester, parents now pay more than £60,000." Very few schools charge £60k/year and those will all be boarding schools.
Public schools have spent a couple of decades upping their game to appeal to the rich. There's been an arms race to get the best sports facilities, theatres etc. Some have borrowed heavily. There's now a tranche of 'all Steinway' schools. (Every piano is a Steinway.) Fees have gone up from 15.k to 60k in 20 yrs. And that's before the VAT thing. Maybe we'll see some schools in trouble? Presumably no more Russian pupils. Maybe no Chinese either? I assume Xi Ping has clamped down on such disloyalty? https://www.ft.com/content/852bf8c5-302b-44ba-a022-79b8ac4fcb42