I think there is a lot of naivety on here.
I was state educated until 16. My parents moved me because I was being bullied due to my academic performance (it was too good).
In my extremely selective, extremely expensive private school I was literally spoon fed. I went from class sizes of 30 to a maximum of 16 (that teacher just about made my Ma laugh out loud when she bemoaned at parents evening how difficult it was to teach such a "large" class).
In my private school people arrived for lessons with the correct books, stationary and generally sat down and shut up ready to learn. At my state school (which is a generally well regarded school in a naive area) the beginning of every lesson was a battle to get started.
Therefore this research comes as little surprise to me. Kids who do well in the state sector (and we are still talking about kids who have done well, just perhaps not achieved all A*)have generally had to overcome the odds to achieve this.
Anecdotally my university experiences also support this research - my flatmate for many years attended one of England's most elite public schools. He had excellent A-levels (4 or 5 As, no A* in my day, but one of those was general studies). He also fucked up every single set of exams we ever had and wasted every holiday studying for resist.
IMO he had become accustomed to being spoon fed at school and lacked the ability to plan and carry out his own work. He and I have pursued the same professional career and the path has been the same - failure to pass our professional exams has prevented him from progressing to the same level as me and the majority of our peers.
Yet on paper when you look at our school grades he is much much better qualified than me.