theoretically, on averave, a given child would perform just as well going to the average state school as they would at the average private school
Theoretically, that is what this article suggests, yes. However, in the real world, no one can possibly think that the same child will achieve exactly the same grades at a school with small class sizes, high expectations and good behaviour as they would at a school where a teacher spends much of a lesson tackling poor behaviour and where lots of children have lower aspiration, bullying those who work hard.
If you’re looking at selective grammars, the difference is likely be much less than it is at a standard large comprehensive.
As the article refers to results going back 8 years, it also doesn’t mention the huge disparities between the two sectors during Covid, or the increasing problem in the state sector in recruiting specialist teachers at secondary level. Those 2 things, of course, will be affecting the chances of children. Many state schools during Covid were offering minimal, or no, work. Most privates were offering a full online timetable from day 1. Of course that will have had an impact on children. A local primary was offering 2 worksheets per week, taking around 10 minutes each. A nearby private was offering a full timetable, including in creative subjects and sport. Of course the children at the private were at an advantage.