I think the type of school isn't necessarily the key. In my experience, the specific school and the specific child are what matters (in my experience). A bright child in a state school that is supportive and encouraging will do well. A not very bright child in an indie school might not. A bright child in a not very good indie, equally, might not do well.
My daughter is in year 6 at a prep school in Bromley. I moved her from a "good" (according to OFSTED) state school (also in Bromley) at the start of year 4. In her case, going to a prep school has been a transformative experience. She ended up scoring well on the Newstead and Kent tests and getting a top 180 score in the Bexley tests. I think it very unlikely that she would have scored so well had she stayed at her last school but that isn't because it was a state school per se - it was because the school was badly managed and there were a lot of internal problems. There is no comparison between the prep school she attends now, and that last school. She is infinitely more confident and the prep school was able support and nurture her talents, whilst her last school barely spotted that she was pretty exceptional (she was bored at her last school and they didn't have the time to really challenge her). As soon as she started her current school, they put her on the gifted and talented programme. Obviously, going from a class of 32 to a class of 14 made a difference, too. For me, it is the best money I have ever spent. I had to scrimp and scrape - and we are by far the poorest family in the year (and probably the school), but I wouldn't change it - and neither would DD. If I'd been able to get her into one of Bromley's well known "outstanding" state schools, however, I expect she would have had as good a result. But you have to pay a premium to live in the catchment of those schools in any case - and they rarely have places coming up "in year" - so, for me, an indie was the best option.
It's worth noting that for the last few years, around half of the year 6 girls at DD's current school have gone to Newstead which is a bit crazy! There is another indie school in Bromley which seems to have around a 100% pass rate for Newstead and Olave's each year! So yes, it can make a difference - but, as I say, there are very bright kids doing well from all different school backgrounds. My advice would be, if your child is happy, keep them where they are and give them some tuition (either DIY or with a tutor) to help familiarise them with the exams. It's very stressful trying to get your child into a grammar school so best of luck and use this forum for advice on the best preparation materials!