I try to divorce my own experience to that which I choose for my DCs but it's hard.
I was privately educated at secondary and in the long run it has done me no favours at all. They only had an eye on their outcomes and I was channelled into higher education that was completely inappropriate for me. I know not all private schools are like this but it makes me very wary of them, particularly when you look at the websites and there's a page just for the leavers' destinations.
I know I could have chosen differently but it's hard not to take your teachers' view of things as gospel when you're 17. My mum just relied on the advice from school. I got great grades but mostly down to my own hard work and was seriously unhappy.
I think I'd have been way happier and better educated I think at a new university. In fact, I know I would have been because I made friends with students on the same course at the ex poly up the road and snuck into some of their lectures. But that wouldn't have looked good for the school. A more open minded approach would have been healthier.
It also made me feel as though there were no other way to do well and that all state schools were not as good. Which, as a secondary state school teacher, I know is a load of rubbish now. I received some dreadful teaching. Because we were all so well behaved lots of the teachers had become lazy.
The facilities were amazing though. And it was a beautiful place to be.
Re. entry to Oxbridge - our local high profile secondary academy is taking a good number of applications for sixth form from independent school students...
So to answer your question, I'd take each school and each child at face value. Ours are at our very average local state school and are very happy. So, for now, that's fine imo.