"DD's state primary which does actually do really well by her is also one that many people go private to avoid."
I think this is a really interesting common thread at primary.
My personal experience could add to either side of the debate.
On the one hand, in a primary where teachers are used to children of widely differing abilities, backgrounds, home lives, languages, then the process of 'adapting to the specific needs of each child' is very ingrained - and this can be to the benefit of the able. In schools where 'the norm' is to be able to teach 'pretty much the same to pretty much everyone', the culture may be 'lazier' in terms of adapting to the needs of a genuine outlier.
On the other hand, DS's first primary, which wasn't great, really WAS a disaster for him, partly because of behaviour, partly because of total lack of peer group.
So although it's an interesting correlation, it isn't predictable.