I agree it depends on where you live - the thing is, where I live, which is also a town with one of the top grammars in the country, the best comp is in a catchment that is for the most part privileged, and increasingly so now the primary is also Outstanding and the catchment for the comp more clearly defined by distance, as more and more families try to buy their way in. The premium on houses locally is big if they're in the catchment. The stats on free school meals (again!) show that low income families are very under-represented indeed. There is also an Outstanding primary a couple of miles away in a very deprived area, but that score's down to value-added. The kids leave at average attainment levels which is wonderful when their relative starting points are considered, but you couldn't call them privileged on that basis, no. And most of their best performing kids wouldn't make it into a grammar unless they had a leg-up, though I'd imagine they might well thrive.
The state/private divide is just too broad a brush for admissions.
I don't really know what the answer is, because if there were enough places for all kids able enough you'd end up with people like me sending them as the competitiveness would be less toxic, and then you'd end up with a two-tier system, with secondary moderns and grammars again. Which was hard as hell on late bloomers. And I don't think we'll ever be able to achieve universal excellence at primary level, at least without the kind of financial investment the Finnish government make. Which is not likely, politically.
Maybe old fashioned streaming, with more movement between the streams and the top streams very academic and pushed indeed, might be an answer, too? No idea how workable that is, or if it's even workable at all, but I'm reluctant to throw so many on the discard pile as not academic, QED, at 11.
My DH is the working class immigrant result of a grammar school, incidentally. Neither parent is that academic, and they certainly aren't privileged at all. But the coaching culture didn't really exist to the same extent when he applied, and his grammar was a lot less competitive than ours because it isn't in a hugely affluent London borough. The results were still stellar, though. Maybe the question should be, why are we not offering that level of education to all our kids, instead of quarreling about a tiny pie available to the lucky few? Aaaaand then we're back at the Finland/taxation point.