Grammar school heads highlighted the role of primary schools but ‘this was difficult when primary schools disagreed with assessment based entry processes and selective education in general’.
^^
Therein lies the the main reason why progress will never be made towards 'fair access' to the Grammars. Our Grammar school has recently introduced another criterion for allocation of places in the event of over-subscription - children in receipt of FSM. All fine and good, but for the fact that due to the following reasons, from the OP's linked post, many many children who would qualify won't even get within a mile of the 11+:
Rightly or wrongly, many heads felt disadvantaged learners had relatively lower educational aspirations.
Disadvantaged parents were sometimes perceived to know less about grammar schools and place less value on the benefits they might confer. And, I would add, be less able to afford the resources necessary to ensure their children were not disadvantaged by not having covered the material included in the often curriculum-based exam.
Heads felt disadvantaged parents ‘often associated grammar schools with tradition, middle class values and elitism’. Parents felt their children ‘might struggle interacting with children from more affluent backgrounds’.
Now, if the 11+ exam were to be run as it used to be, and not as the opt in procedure it has become nowadays (i.e. every child takes it, unless they opt out) then this would mean the primary schools being involved once again in administering the exam. Maybe this would encourage the schools to actually take an interest in the outcomes, and equip the children with the tools and knowledge they need to attempt the exam, rather than leaving it to paid tutors or savvy parents.
Perhaps then there would be some hope of reaching the original target market of the Grammars - those bright children who often don't get the chance to opt in to the process as it stands now, because circumstances, or their parents, have effectively opted them out long before the exam takes place.