well, ds gets FSM. he has passed his 11+ and is pretty certain of a place at Grammar School.
We are in a superselective area, and we self prepared for 11+, as we couldn't afford a tutor. Then for July and September (exam early Oct) my Mum paid for him to go to a tutor, this was mainly because my mum was in hospital and I was massively supporting them, and I couldn't do both. The tutor did, pretty much, what I was doing.
BUT, I am a highly educated person, our family is very vocal. We read, talk, discuss and debate endlessly. My dcs spoken vocab is very wide and they read loads, so have also a wide reading vocab. Those are mostly things they get from the family environment, not from tutoring. Education is also very important for our family, and we support it, prioritise it.
I don't think that most kids receiving FSM have those family/background advantages. It is not just about tutoring or buying past papers, it is about the whole family support package. While many families want to support their kids, their don't always have the skills to do so.
Sadly, I think the gap is already evident at reception (vocabulary, familiarity with books and story structure, familiarity with counting/number games) and just widens as the years go on.