I decided not to send my kids private for a few reasons. Firstly, social - why should the tax payer subsidise school fees by paying the VAT, making it cheaper for parents that are already wealthy? I don't think there is any way my conscience could square that circle. Yes, I know the w word is controversial but if you pay for something that you can get for free, even if you have to tighten your belt, you are wealthy.
Secondly, I am a teacher and I cannot see that there's anything to be gained unless your local school is feral. Teachers in the independent sector accept poorer pay because, quite simply, many of them cannot teach. Yes, they deliver academic information but it is pretty much one size fits all, especially if the kids have to pass an exam to get a place. Many are not qualified teachers and rely on the largely compliant nature of the children. I don't see this as VFM.
My nieces and nephews go to a fairly prestigious school (brother in law earns 280k plus bonus, so go figure) and, to begin with, they were ahead of my kids in measurables such as maths and English, they were even learning French, but this is because they were hothoused and this wears off as progress naturally plateaus. There's nothing sadder than seeing a 7 year old completing masses of homework on a sunny Saturday. The enrichment activities were invasive with the kids having clubs on Friday evening, Saturday afternoon and Sunday mornings. In the long run, my kids did better than their cousins at GCSE and A Level so I'm not sure it was money well spent. I also scratched my head at my rels joining a little parenting club to buy additional tuition for their kids in the run up to exams.
Class sizes are always touted but the maths is revealing. In simple and artificial terms, 1 teacher and 30 kids for 1 hour is 2 minutes per kid. 1 teacher and 12 kids for 1 hour is 2 minutes and 30 seconds per kid. Do larger classes make any difference? They make nervous kids less visible to their classmates. They facilitate healthy discussion. There's always someone in the room who'll help you.
Of my three nephews and nieces, one is resentful that he wasn't allowed to attend school with his mates and play for the local sports teams, he feels he has no social circle. One has had a bit of a shock in terms of the lack of deference from her uni teachers. She has had to learn very quickly that a deadline and associated sanctions happen, something she never experienced at school as the teachers run scared of the parents and are treated as 'staff'. The third is most well adjusted as she is the youngest and her parents are trying to undo their past errors by building a social life with the local kids and limiting her to one weekend club.
It's all horses for courses but, unless it's an elite school, I don't think there's much to be gained by going private.