Interesting - I wonder where you get your information? Sounds as though you may be a medic yourself? Have you engaged in the private school community?
5 doctors in my two kids' year-groups alone, near Southampton, of which I know 3 do mainly/entirely NHS work. 3 work FT to pay the fees and it's marginal; they might keep working but they really fancy 3 days/week. All are in dual-income families where both are higher-rate taxpayers.
In the parents' campaign we also have dozens of nurses and other NHS workers (physios, radiologists, you name it) on lower wages. Typically they are the second earners; it's the joint income that makes independent school possible, but it's also the joint FT work that puts a strain on the family. Who forgot to unload the laundry? etc.
so (1) counter to the millionaire stereotype, these are normal families with well-paid, unspectacular, demanding, useful jobs (2) take away the fees and it gives them new options.
You're absolutely right about the tax rates and disincentives over £100k, plus childcare cliff-edges which are bonkers. You need a real, serious reason to work in that range. Do you see the inconsistency between (1) "school fees are expensive and out-of-reach for most" and (2) "everyone who can afford school fees is so rich they have money to spare"?
Childcare: our day school does 0745-1830 every weekday included in the fees, so that's quite a help with the wraparound care. Admittedly leaves you with the longer holidays. Just one of the many perversions of this policy: what serves as 2 hours childcare for us (the timetable finishes 1625) will become VATable, whereas after-school clubs at state schools won't be. So they say.