I haven't read the full thread but I have read the OP's posts. There are various options.
I got a council assisted place to an independent school. I really hated it at the time (snobbish and most of the other pupils had little in common with me, in various ways), but, in retrospect, I wish I had had the grace to be more appreciative of the teachers. One of the issues with music scholarships etc I think is that the school might need a child happy to conform in some ways - wouldn't have suited our particular music enthusiast, I think.
There are outstanding state schools - like Brampton Manor- which are high achieving despite being in a tough, London environment.
There are very high performing sixth forms in south London, especially if you have a child who is very gifted in Maths. There are also massive sixth forms like Hills Road and Peter Symonds that can offer children a lot of options, and have fantastic Oxbridge and Russell Group acceptance rates.
And there are really excellent state grammars.
We deliberately moved to a village in an area which would be lovely to bring up children. There are also good schools locally. My husband laughed at me checking the Ofsted of local state schools before we bought our house 18 years ago, but it is great that our kids have had such good options. Mind you, the schools could potentially have changed a lot for the worse over that time, we were lucky.
We have done state all the way.
If you have a property in London, then you might well have a lot of options if you wanted to move out, or move to a grammar area.
We have also done a lot of volunteering with the kids' schools, and were lucky that the schools had a culture of like-minded parent volunteering.
I would be happy to DM you, but I'm not sure what happened to the DMs.