To me there is a huge elephant in the room when it comes to poor kids and grammar entrance.
Take 2 children who are bright. Raise one in a household where there is loads of conversation, complex vocabulary, lots of reading, both books being read aloud and models of adult reading. Throw in dinner table discussion about politics, the world, life the universe and everything. Give the child opportunities to visit places on holiday, museums etc etc etc.
Take the second child, and put them in a very low income household with parents who have low educational attainment themselves, and where no-one really reads for pleasure beyond magazines.
Guess which one will have the verbal skills and vocab to pass the 11+
If you want to improve the number of kids on FSM entering grammar school, you need to give them access to wider educational opportunites earlier, like at age 7.
(yes I know that is a stereotype. My kids are on FSM and got in, and I am articulate etc etc. But when I look at the other bright kids in dd1s class, who could have potentially passed the 11+, they were from families like that, and they would not have passed, and it would not have occurred to their parents to enter them either, but they had the potential)