My kids are still young, so I'm not speaking from any of that sort of experience. But I just wanted to share an experience of a friend's...
Her son didn't pass/sit the 11+ test and ended up at the 'best' non-grammar in the city. In our city, we have 3 grammar schools (of course, you can travel and go beyond the city... ). 2 single sex schools and 1 mixed.
He's currently in his second year of secondary school and he recently recorded a class lesson on his phone (as he was previously complaining that he couldn't concentrate, focus, learn...) and it turns out that it was like a zoo, with the teacher doing nothing more than crowd control.
His parents are now considering private ed. He's not the most academic, but he's very much talented in other things (sports, music etc). I don't really know what his options really are, but I thought, when I heard this story, that the system's somehow failed him and his parents...
I guess what I'm taking away from this is that going to grammar school does not mean you automatically get to do medicine at Oxbridge, but going to grammar school could significantly enhance your chances because the environment that you study in is perhaps more geared towards learning (a combination of course, of the school curriculum, teachers, headteacher's leadership, other students and their attitudes....etc). Of course, I'm talking with reference to one case example.... I can't comment on others.
If I were you, I would encourage my child to sit the 11+ and go for the grammar school.