First, in case people are interested in Warwick they are holding some subject information evenings in London at the end of this month. See link: warwick.ac.uk/about/london/study/pre-university/subject-information-evenings/
Secondly, my DS school (also a super selective which I assume BPIsok's DD's school may also be) has a similar view to theirs. It is however a school which got 95% 9-7 A*/As at gcse this year.
It had 50 Oxbridge places. A large number of people apply because they superselect ie. fall within the top 5% so if at another school they would be in the top end but at their school it is "average"!!
DS would not be considered the "genius" end of his cohort at school but will still apply as he is one of the better ones at humanities (and wants to study History). The school has a very large Maths/Science uptake at A level (probably at least 75% of cohort) with Economics being the next largest.
However they will not block anyone who wants to apply but they do offer support and suggestions for super curricular reading, volunteering etc. which possibly some state schools either can't or don't offer. BUT Cambridge/Oxford do know that schools like DS's does have this in place and I suspect it is the case of them having to outperform others in that position to get their place.
I think the main thing is that an application to Oxbridge is more normalised in that setting than a state school except if one of the well known high performers such as Peter symonds, Farnborough, Hills Road etc.