@WiseLemonFox I have a daughter who has gone through Sydenham Girls (the state one), with GCSEs all at 7-9 and now loving the sixth form in Year 13.
I also, coincidentally, have a Year 11 child at a private school - though not a local one. She's got a government music place at a boarding school. She moved in year 10 so has experienced both Sydenham Girls and the independent sector.
DD2's view, having experienced both sides, is that she wouldn't pay for the teaching at a private school just for the sake of it as she thinks it is definitely not better.
There are nice things - small classes, and a swimming pool, and lots of text books to go round, but in general moving made her realise just HOW good sydenham was, particularly for maths and science - where she found the transition extremely easy - so much for state school children being behind.
DD2 is high ability - more so than her sister, who is bright enough but had to work for her grades. Sydenham stretched her with lots of extras and is now offering further maths as well as triple science to the top of the cohort. I think she would have got great GCSEs there. DD1 did a lot of debate which really helped her confidence and is a strength of the school.
Friends of mine who have had kids at Sydenham high are ok with it - but not wowed. Many move for sixth form as it is small...JAGS is a different ball game - more academic, high-quality music etc.
My caveat is that if your child is very musical and wants that as a career, don't go for Sydenham Girls at the moment as it has become a bit of a weak point, which is why DD2 left.
If she hadn't been offered her dream school with a massive government chunk of funding, we'd have moved her to Prendergast, where a space opened up for her.
Otherwise, Sydenham is a great place to be.