It's not just that children who have FSMs don't get in (i.e. don't pass the test), fewer apply in the first place. We live in an area with only a few grammar schools (i.e. not fully selective like Kent) and I have lost count of the times I've had conversations with parents of bright children who get FSM (e.g. single parents on low income) who could have tried for the grammar school but don't want to- they don't want them to travel, want them to go with their friends, perhaps lack understanding of how a grammar education would benefit them. This is a poverty of aspiration issue and not about tutoring.
Conversely, we are friends with lots of immigrant families who don't have great jobs in the UK and they fight hard to get their children into the grammars, even if they have a higher chance of failing the exam due to only being in the country for a short time. They know an excellent education is a way for social mobility if you are an immigrant without a profession themselves, and so they are prepared to sit the tests, pay for tutors (whilst working long hours in crappy jobs themselves) and pay for travel to the schools which is often quite expensive.
I think the social mobility has got worse as everyone doesn't sit the 11 plus. My dad and his brother both got into grammar schools because everyone in the whole town sat the entrance. My grandparents left school at 13, worked in trades and my grandmother was semi-literate. These days they would probably not bother, because it would depend on them to push their children forward, which they would not do. When it was part of the system, there was more of an equal chance of poorer bright kids coming forward.
It was also not true that there was a huge segregation between technical skills/academic ones, my father went to grammar but did well in what is now called design and technology and was an excellent craftsman, so had more choices of career open. The curriculum is quite narrow now.
I went to a 'bog standard' (as in crappy) comp and there was a lot of bullying, disruption and general picking on academic children which made life quite unpleasant, some of my friends have really not got over their school experiences. I would like to say this has all changed, I fully believe people when they say the academic standards are better, but am unsure the behaviour has improved or that it's easier to be clever.