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Yes, some schools really go for it. My old (state) 6th form college was usually #3 in the list of Oxbridge entrants behind Westminster and Eton.
There was a big push within to get people to go for it and I resisted successfully. (Out of the 15 people in my Further Maths class 12 went to Oxbridge to do Medicine, one went to King's College London to do Medicine and me and one other did Comp Sci [Sheffield and Leeds respectively].)
That is great - judging by their 'Oxbridge Progression' wooden board they've been making this a focus for a few years now. Not that that is wrong of course, on the contrary: from what I've heard, the problem for a lot of underprivileged pupils is that they go to a school which doesn't provide any proper support for an Oxbridge application. So the thought barely comes up, and if it does, there might not be anyone with direct experience or at least useful knowledge to help them along with it. This school on the other hand puts it up there as an aspirational goal, which I'm sure helps a lot.