It is a lovely illusion, isn't it, that if pushy middle class parents sent their children to state schools they would do better.
In fact the majority of pushy middle class parents do send their children to state schools. Less than 10% of children go to private schools, yet 25% of working people fall into social classes AB, and 50%, into the usual middle class definition that covers social groups ABC1.
The problem is that there is generally a correlation between the socio-economic background of the catchment area of a school and the quality of the school and since those that own homes tend to be better off than those that do not, and home owners also buy the best house they can afford, schools with lots of 'pushy middle class parents, tend not to be in the areas where state schools are most in need of improvement.
My DGC are attending excellent state schools. In fact nearly all the schools in their city seem to be excellent - and that is because over 60% of the population fall into socio-economic groups ABC1 so 'pushy middle-class parents form the majority in most schools.
But how much is the excellence the socioeconomic group of their families in purely money and occupation terms and how much in the higher educational levels of the parents, that means they are likely to be providing a culturally enriched back ground for their children: books available and brought into the family, talking to children more, reading to them, taking them to visit places etc, walks in parks, visiting (free) local museums and libraries. Things that are available to everybody regardless of social class, but are known to be used most by the better educated.