The unfortunate truth is, private schools exist for a reason: because they give you an advantage.
This isn't due to Oxford necessarily: as has been said, the official line is that two applicants of the same standard, one state one private, the state has the advantage.
But there is a lot that goes into getting to that point in the first place. You say ignore grades... why? Its a vital factor and one that objectively changes with type of school. And then there are factors like interview coaching, advice on making a solid application, and arguably just personal development that is gained by being in such an environment.
The results are fairly plain to see: 45% independent intake, with the private schooled having a slightly higher success rate when applying in addition.
There is possibly an argument for someone who is super confident of their grades and super confident that they won't benefit from any kind of coaching... then maybe state would be no different. And then of course there are good state schools, including grammars, and occasional lower achieving private schools, but my opinion is that these are a minority of cases. Private schools exist for a reason.
My aim here isn't to discourage state-grammar or state-comp applicants. I was one and my time here has been great. The difference is small and state applicants still make up the majority of successful applications, forming a greater proportion each year at the moment. I just think that its worth acknowledging that our current society is structured such that those who can afford it can and do advantage their kids, for good or for bad.