Sorry to hear that Metgala's daughter had a bad time at City. As a parent myself just wanted to address a few points:
I don't think the Head invited the Duchess of York. The Duchess of York invited herself as she was patron of Street Child at the time. City girls have raised £200k for Street Child. My daughter said her speech was pretty barking.
Though I have not kept a tally, it is not my impression that there has been an unusually high turnover at City as a result of new head - leavers and joiners seem similar to those at my ds's school. There were a few high profile staff leave last year, but mostly because they went to bigger jobs than they could find internally not because unhappy, e.g. the Head of Maths became Head of Kings College Maths School, Head of Music Deputy Head somewhere, the Head of Drama moved to Westminster. But there are lots of staff that have been there a while and lots of stability at the top (I think the deputy heads have been there a good while for instance) and we are impressed with new head of maths.
My dd has not had tutoring since being at City but I know quite a few do maths and science, I am not sure that this is much different from other competitive private schools? Maths is the most popular A-level at City by far so it might reflect some people taking it who are not that naturally gifted at it? I do get the impression that there is a bit of peer/parental pressure to do maths at City.
The school is woke, I think mostly positively so, but I am sure that there is a bit of stuff that would annoy the more free-speech liberal side of me. But it's in the Barbican and a lot of parents from Islington so would expect it to be somewhat woke? I don't get the impression though that it is uber woke.
As for VFM, I am dubious about the value of private schools over good state schools (I think a lot get good results because they get good kids) though I think they do make a difference and prob partic in subjects like maths where state schools can struggle to get good teachers. I think that the pastoral side can be better at private schools (though obv being at an academic school like City can bring new pressures). So I think whether it is a good choice does prob depend on how much of a sacrifice it entails and whether there are good alternative state options available.