My understanding was that private schools opted for iGCSEs years ago at a point when there was constant chopping and changing, and lots of course work that could be manipulated.
IGCSEs were initially designed for overseas students. There are some quirks - no Shakespeare in the English Lit as it is not very relevant for a kid at an international school in the Gulf. My observation was that schools at the time were mainly interested in finding syllabuses that provided a good preparation for A level study. Some may be easier but I am sure the main attraction is the relative stability. Independent schools are independent and they can choose to stay out of English education politics.
There is now a continuing, slow and steady shift towards pre-U and IB rather than A level. Again because academic private schools are mainly interested in preparing their pupils for University level study. (And before someone whines there is also a continuing trend for those pupils to look outside the UK for University.) With many private schools still insisting that the majority of pupils take four A levels.
I almost feel sorry for state school pupils, having to share lecture rooms with private school pupils who clearly have no right to be there. I hope they float in on their moral high ground and ensure that their superiority is immediately obvious.