Private is most definitely not always best. Some private schools are terrible. They have teachers who wouldn't get jobs in the state sector. Private schools can be money for old rope - often are.
But again, OP is not talking about the independent sector in general. She is talking about a SPECIFIC super-selective that achieves 90% 7-9 at GCSE, versus a non-selective where that figure is 20%.
That is a HUGE difference, whichever way you look at it.
I think some people don't realise how competitive it is to get into that type of school - whether it's a super-selective grammar, independent or academy.
The teachers are no better. There are good and bad teachers in any school. But the pace is totally different. Someone working at a 7 in a super-selective will be seen as 'struggling.' In the lowest Maths sets, they will be getting 8s.
When teachers can just walk into a classroom and everyone is in the top probably 2% of ability, they aren't worrying that anybody isn't keeping up. They finish the GCSE curriculum in Year 9 or 10 and the rest of the time is just consolidating.
There are super-selectives (private and state) in inner London where 15 children sit for every place - eg. There is a school near me where over 1,500 sit exams over maybe 2 days for 100 places at 11 plus. Even after that, there are interviews. It's a stressful process and many are sitting for 5,6,7 schools. Children at the very top of their primaries will often not get in. There are no guarantees for even the brightest child. Numbers-wise, it's more competitive entry than Oxbridge., No wonder, of those who do get in, nearly everyone comes out with 8s and 9s.
This school OP is considering sounds like it's outer London (because she mentions coaches), so maybe it's Hampton, Dulwich College or Merchant Taylor's, something like that? They bus them in from wide catchments. These schools are also very competitive entry. There will be waiting lists of people phoning the schools every day to see if any places have been turned down.