Just think of the cost of sending even one child right through private school. If you want to send them somewhere swanky, let's say Eton, that's about 30K a year, before extras, so well over 200K in total. Just for secondary.
Or a nice enough local private school. Near me they start at about 10K a year. A little less for lower years, so from year 3 onwards you are talking about about 100K, before music lessons, uniforms, school trips, all the rest of it.
That's a lot to pay for...say 4As at A-level, if it's educational attainment you're looking at. I don't think that's value for money, especially compared with the baseline of a good state school education.
Spending that 100K on carefully chosen one-to-one private tuition? Led by your - and increasingly your child's - interests and priorities. Especially effective in the areas that almost all schools - including private schools - do badly, such as languages. Or language immersion trips when older. Tutors don't just have to support the school curriculum either, they can develop individual guided study programs with students.
Or speaking of when they are older - bear in mind what higher education might cost once your DC gets to 18. As of now, it will cost a student around 40K to get through a 3 year degree. 80K for students of long courses, like architecture. And as the government are going to sell the student loan book, we can now expect these to be repaid at commercial rates.
In previous decades the time critical periods for young people's career prospects were GCSEs and A-Levels. Increasingly, that's no longer the case, and the most crucial (and financially testing) periods of young people's lives are their years in further/higher education and the post-graduation period.
It may not be that their years in primary and secondary education are the best ones to throw money at. Of course good foundations are incredibly important, but I believe, and most research seems to indicate, that there are established at home, through parent-led example. Personally, if money is a concern, I'd consider setting aside a bit more time for educational activities at home, if you can. Supplementing a decent stateschool education with well targeted tutoring, and thinking about what the hell the higher education will look like ten years from now.
But as I said, if you want to buy a certain peer group, then private school seems to work well for that. That's its entire purpose, as far as I can see.