I was at QAS, there was absolutely nothing snobbish (and very little posh) about it, and I have very happy memories of my 7 years there. I am still in touch with numerous friends, and got straight A/Bs all through and went to Edinburgh university. I did, btw, also get into the Abbey but my parents preferred QAS. 90% of my year group went to uni so I don't think you could really say we weren't bright.
admittedly that is all rather in the past now ! But I do know a couple of people with dds there and they are extremely happy with the school.
i have no experience of Headington so can't comment.