Ok - to understand how Oxbridge works you need to understand the Collegiate system.
If you apply to (say) Caius College Cambridge to study MML , you are not one of 1000 kids applying to Cambridge trying to get 150 places. You are one of 24 kids who applied to Caius to try and get one of the 4 places.
Once the four get there the same admission tutors will have to teach them for 3 years before they sit their finals and the admission tutors / personal tutors are judged on the results.
So what they care about is getting the most able four. And they invest huge amounts of time and effort in working out who those four are.
They know that someone from Eton who performs equally to someone from a sink comprehensive in the midlands is actually far less able.
They also know that a school like Tiffin is far closer to Eton than it is to a sink comprehensive in the midlands.
Oxbridge academics are, by definition, really quite clever. They know how people try and game the system.
There is no guaranteed way to get your child into Oxbridge except by him being of the correct level of ability and having a true love for the subject.
The last time I checked - it is still believed that proportionately there are far more highly able state school kids not going to Oxbridge than highly able private school kids. However, a big part of that is that too many state school kids don’t apply. The admission tutors are pretty good at getting the best of the applicants.