In fairness the Oratory is a specifically Catholic school with a mixture of students. My nephews went there and had to undergo a very strict faith-based assessment to get in - they certainly don't live in a very expensive house, but do come from a deeply devout family who have given a lot of service to the Church.
I don't think I'd characterise it as hypocrisy that Starmer's children are in good state schools. Most politicians are relatively wealthy and will live in relatively wealthy areas, and like all parents want their children to be well educated. In terms of 'gaming' any system, the charitable status claimed by private schools is far more unseemly.
There can be a pretty derogatory tone from private school parents on MN about state schools in general, and private school parents have done themselves no favours in saying that they're motivated by wanting the best for their children, or that they've made sacrifices - inevitably that suggests parents with children in state schools care less about their children or haven't sacrificed enough.
Choosing a state school can also mean a strong commitment to the local area and community, wanting your children to mix with a broad range of people, and down to the character of the child - if they're studious and motivated and you're involved with their learning, that can go a long way and there may be other advantages such as (eg) Oxbridge colleges trying to widen their admissions for more state educated entrants.
When you look at Ofsted's stats, of the 3,332 secondary schools inspected, 557 are ranked outstanding with 2,122 rated as good. That's a pretty high proportion (c.80% either good or outstanding). I think that's worth keeping in mind if people fear that moving their child to a state school will inevitably be a bad move, particularly if they reflect on the money they'll save on fees that can be used for tutoring or otherwise augmenting their children's education.