When you say all the local secondary schools are ‘quite rubbish’ - what are you basing this on? Have you visited them?
We decided to stay, because we really like living in London, and when we looked at possible towns to move to in Kent, Surrey etc to be honest, I just didn’t feel at home there. Also we didn’t want to face the grammar school stress / lottery. DH didn’t want a long commute to work.
The local secondary schools near us in SE London aren’t terrible, but they can’t compare to grammars or private schools. However, they do have benefits - some very good teachers (often young, enthusiastic teachers, rather than someone who’s hanging on for their pension). Intake is much wider, so exam results are never as good as grammars, but if your child is motivated, they can do very well.
We also decided we’d spend the money we save on not going private on Tutors where necessary.
However, I’m not a Tiger mum. You’ll find plenty of mums on here who would do absolutely anything rather than send their child to the local LA secondary school. Personally, I like to look at the bigger picture of what’s best for the family overall.
My DD did well in her GCSEs, now doing A Levels. She’s happy, enjoys the freedom and culture of living in London, and she’s mixing with people from a wide variety of backgrounds which can only be beneficial to her.
But hard to say what’s best for your family!