It really depends on what you want.
My DC1 got 3 x A* at A level and should get a 1st at uni this year from going to a bog standard comp and a selective state sixth form. He knows the value of money, is hard working and has mixed with DC from all sorts of backgrounds. This is something that I value. He didn’t get into Oxbridge, though, which he might have from a private school but is at a RG university.
My DC2 has SN so would never have been accepted into a private school. Luckily the state comp he went to valued him as an individual and he did fairly well with an EHCP. He’s currently doing an HND at college.
My DC3 is a bit lazy. He will do reasonably well at school, but would definitely have benefitted academically from being in a private school. He could do with the extra push that private schools give wrt doing his homework and revision. He doesn’t choose to get involved with many extra curricular activities beyond scouts and music and may have had more opportunities within a private school. But he would have missed out on the mixing with all groups of society, and if I could afford it, he could have attended more activities outside school.
If your DC are clever and have a good work ethic they’ll do well in the state system. Maybe not quite as well as they would in a private school but there are other benefits, not least the cost. If they are reasonably bright but need some ‘encouragement’ to work hard, a private school would certainly help. If they have SNs IME, the state system is better. There are private special schools but few and far between.