I just wanted to add another dimension if I might.
I have two boys who are both at Pates. They have been to 3 different state primary schools (Northern Ireland, Scotland and England, we didn't get round to Wales!!) due to my husbands job. I would not say they are exceptionally bright but they work hard and have a good attitude to education.
I thought that most of the primary schools they attended did not exactly stretch them but I used to ensure they went the extra mile in terms of the quality of their homework (no shoddy bits of paper) and gave them extra extension exercises at home. I appreciate not everyone can do that but if you can it really helps you to understand how your child learns, their strengths and areas they need to improve. I also tutored my kids for the grammar school test, and believe me, it is not rocket science but does take time and dedication.
I cannot comment on the private sector as I have no experience. I would just say, it is a lot of money!! Pates can be achieved without a private education, many many kids have done it.