I think one of the explanations for the difference in Oxbridge numbers, is the differing numbers of teachers from Oxbridge in the 2 types of school.
Both have teachers who have been at Oxbridge, but schools like Westminster have a very high percentage of Oxbridge educated staff. They are 'insiders' of the system and there is a very smooth Oxbridge machine running, with personal connections to many colleges, and a very developed Oxbridge programme.
Grammar schools, I think have lower Oxbridge educated teacher percentages. There will be many in those schools from RG type Unis too. For them perhaps, Oxbridge is something they are less familiar with and less committed to. They see value in the RG Unis too, that they went to.....so there is just less of an QOxbridge culture. There will be an Oxbridge programme for the candidates, but probably not in such a developed way. I really think it is this, rather than the fact children are less clever....they are often cleverer. The expectation, on the part of the kids, the parents and the staff is perhaps less.
Equally, I expect if we were to compare the %of parents with Oxbridge degrees, we would find more in the top independent schools. This too feeds into a culture of it being the norm, what one expects to happen etc.
This also explains why many Comps have a history of never sending anyone to Oxbridge, despite the fact they may have a handful of kids who are up to it. Many Comps have absolutely no Oxbridge educated teachers on their staff. Although the Oxbridge outreach programmes try to widen access, kids in these schools just don't have access to staff with expertise about the application process. This means they don't tend to even apply and if they do, they are often not as well prepared.