Hi OP, You ask about 'choosing' between a grammar and independent school to 'game' Oxbridge chances. Well, only a handful of boroughs even have the grammar system so, for the rest of the U.K., this is a non-issue.
On independent schools - all that happens is, an applicant's grades are contextualised within the average grades of THEIR cohort in THEIR school. So, somebody with ten 9s applying from a super-selective independent school such as Westminster, may get a contextualised score of '0' because this is the average grade profile in that school. On the other hand, if they were applying from 'sleepy unselective independent school in the middle of nowhere,' the same applicant might get a stronger contextualised GCSE score because the average grade profile in that school might be 6666666666 or whatever.
Exactly the same happens for grammars - they know that some grammars are much more academic than others.
Exactly the same happens for all other type of state schools, They know full well that somewhere like Brampton Manor is hardly the same as your average comp, let alone a comp in a deprived or 'low access to HE' area - so they contextualise accordingly.
Nobody is trying to squeeze out 'private school kids.' But broadly speaking, around 28% of all A / A star grades achieved at A-level are achieved in the private sector. So all being equal, the numbers of offers made should reflect that. And they broadly do - at Cambridge about 25% of students are from the private sector.
Look at 'offer rates' on the Cambridge website. These are pretty much the same across all sectors - independent schools, grammar schools, comprehensive schools and sixth form colleges. The only schools with a slightly lower offer rate are FE colleges.
So no drama and it's the same contextualisation process, regardless of school sector.