Purple
I was sent to private school by loving parents who believed education was the best thing they could give us and sacrificed a lot.
I ended up feeling guilty about their sacrifice.and when I got a bad report they used it as a stick to beat me with which I felt was unfair given it wasn't my choice.
I hated being the poorest.
I'd say, if the local school is good, go for it and tutor any deficiencies. You are in my view right to be concerned about social deficiencies. Given the way that private school fees are going and uncertainty in the economy, don't put yourselves under unnecessary stress. And , it can be easier to transfer to a private school later than to a good state school, if you want to make that choice. And HE is widening participation.
We did ultimately send our DC private, so you may think this is hypocritical if me. But he started off in what was supposed to be a good state primary. They couldn't cope with his learning needs, his rare medical condition and they did nothing about him being bullied as a result of those issues. He applied to the local state schools but the nearest was at that time the LA's sink school and we didn't think he'd survive. But neighbours who managed to work the system got their children in to the excellent other local state schools (catchment area changes) where they have done superbly.
Given a different child or a different local school or the same one but as it is now, I'd have no hesitation.