bilbo as an admissions bid and also a parent of two DC currently at a highly selective public school, I still question the efficacy of the Oxbridge prep provided.
I'm not disputing it has any impact, but less than you would think.
What has far more impact is the quality of education generally. It is serious, rigorous and broad. It prepares pupils for academic life beyond school which of course includes Oxbridge.
If you made an active decision for your DC not to receive that type of education then you can hardly bemoan that it didn't include Oxbridge prep.
As for contextual offers, they are rare but I can assure they are made from time to time. However, care must be taken as in the past some students in receipt of lower offers have not fared well.
In practice, what happens is that contextualisation takes place during the admissions process.
Applicants from certain schools are held to a higher standard in terms of their results, predictions, performance at interview and pre test.
Applicants flagged for contextualisation will have it taken into account. Of course this does not imply an easy ride.
As for outreach. Both universities do loads. I personally have visited lots and lots of schools. I have ongoing links with any that want that.
However, we can only offer the horse water. Some schools decline visits. Some ignore us altogether. And some that do invite us for visits do not take on board any of the advice we offer.
Not all of course. Some are absolutely committed to widening participation.
So what counts?
Good GCSEs, yes of course.
Good AS - who knows? As they become optional, can they be any sort of measure?
Pre tests - currently very important at an Oxbridge college near you.
Personal statement. Don't overthink it. Say why you want to study this subject, then back that statement up with some evidence from things you've experienced/read etc.
References are indeed worth checking. This is a good place to point out any flags for contextualisation, and also a good place to signpost an specific academic achievement.