Admissions tutors are not daft: they can see from the UCAS form whether the bulk of a candidate's education has been in the private or state sector. A sudden switch to the state that cannot be explained by reasons of subject choice, desire for a change, etc, would raise cynical questions, and would not enable the candidate to be seen in the same way as a student educated comprehensively throughout.
Similarly, there are state schools and state schools: pupils from St Swotty's Shireshire Grammar are not regarded as having had the same life chances as Innercity Sink pupils.
There are as many myths, sour grapes and misconceptions in independent schools about Oxbridge entrance as there are misconceptions in the state sector.
Quite a lot of pupils feel discriminated against, simply because previous generations of alumni sailed into Oxbridge unhindered. It has to be pointed out to them that when they achieve the same grades as a candidate from a disadvantaged state school, objectively speaking the latter will be the stronger candidate, simply because they have had to overcome more hurdles to achieve them, including larger classes, poorer resources, less well-resourced teaching. Taking the state school pupil is not discrimination; that's wise selection.
Similarly, it has to be pointed out that less-than-outstanding GCSE or A level grades when you have had gerzillions spent on your education in as close to ideal surroundings as can be created, means that you are either (a) not hugely motivated or (b) simply not very bright. Either way, you are not that attractive to an Oxbridge College. Some of them grasp this; others find these truths too unpalatable to consider.
I teach in an independent school; the state educated me (and will educate my children) and I didn't go to Oxbridge, but prepare students for prospective entrance.