This research is about Australia, where something like 40% of children are not in government schools. The socio-economic/class associations of private education are not a good match for the situation in the UK. Basically, Catholic schools are part of the private system, but don't for the most part select.
Do the maths; with this level of buy-in, it would be amazing if all of those students did better than the government schools
There is good deal of cachet attached to private education in Australia, but this does not always translate into academic achievement, because the parental support/ attitudes to value of education is not always there. In a nutshell; lots of dosh but not much going on between the ears. It's about the look of the thing. Successive governments have done an amazing job of making private education the default position if you really care about your child.
I've met private school teachers and it's as if they're owned by the parents. The mindset is not "I've purchased the means for my child to have a better education, rather I have bought the education and teacher had better make sure my child gets it".
On another note, the scale of government funding for private schools would make the UK private sector weep with envy.
Funding for government schools 2013-14 - $4,448m
Funding for non-government (private) schools 2013-14 $8,736m.