I'm in London (not central). I have chosen to send my child to a highly selective independent school in September (with a bursary, could not have afforded it otherwise).
For me this was because although I have in theory lots of good schools near me which could meet my child's needs, in actual practice I have only the school that will be allocated to me on offer day and maybe another couple on waiting lists, with all the resulting uncertainty and disruption. We have no more choice than someone in a small town with a few schools in striking distance, perhaps less because of density of population and shortage of school places. The school we will be allocated on offer day is fine, yes. But it doesn't suit my child for a number of reasons and I don't want her to go there. None of the state schools I applied to do banding or aptitude tests apart from the super-selective grammar, which she may or may not get into (and which we are not accepting, following the bursary offer).
The school I would prefer out of the state options is single sex so some selection (and postcode selection) but it is otherwise just a normal comprehensive. I would be delighted if my child could go to this school but we would have an anxious summer on a waiting list, ringing the admissions people every week to check what was happening and it would not be a happy time but rather a stressful one - and there might be no place at the end of it.
DD got offered a bursary. She will be going to a school that offers stuff that she will greatly benefit from and is interested in and she will no longer be subject to the national curriculum or the stress of Ofsted which I think does schools no good at all. I'm sorry she could not go to the state school I liked but I'm not sorry that I have prioritised her wellbeing and future academic enjoyment over anything else. I'm NOT choosing 'people like us' as actually her primary school is highly socially mixed compared to other local ones and I did choose it as my first choice in full knowledge of that; the social mix doesn't matter to me. I just want her to go somewhere that suits her.
London schools have been funded well because until not that long ago they were doing badly. They were doing badly because of the inevitable inner city challenges which are greater in London than elsewhere and because London suffers considerable population churn, more than most other cities in the UK. The constant movement in and out of schools by pupils did mean that London schools were having to work harder than most just to stand still. The extra funding helped. But saying 'Oh London gets too much' is NOT the point. If all schools were funded to this level, I cannot help but think that society would be a lot better for it.
The progress scores and attainment scores are largely meaningless for very selective schools - if you have a cohort that has achieved 90%+ A and A or 7-9 at GCSE and they go on to achieve 90%+ A or A at A Level then they have made expected progress. This does not take account of the extra stuff that is offered by a really good school, whether state or not.
The funding thing is key. We all need properly funded schools for our children and it is ridiculous that we cannot achieve this in a country that is one of the richest in the world. Yet for some reason the majority of the country elected a government that absolutely will not deliver this. Go figure.