For what it's worth (not a lot!) my view on 'what a kid needs' to prepare for the 11+:
- Familiarisation with the test format (in particular, how the answer sheets are structured)
- Familiarisation with the test question types (NVR/VR/Maths/Comprehension/Cloze)
- Feel for the speed at which questions must be answered
- 'Exam technique' - skipping questions that are taking too long to answer, checking answers with any spare time at the end, eliminating wrong answers, etc etc
- As large a vocabulary as possible
- Help with the maths curriculum that will be tested (particularly algebra if our primary school is anything to go by)
All bar the last one are things that can be done at home, without a tutor, if your child is ok with taking constructive criticism from the parent. Depending on the parent's maths knowledge, the last one may not need a tutor either.
And, of course, your child needs to have the right sort of talents that the grammars are sieving the cohort for - pattern spotting, etc. Not much a tutor can do about that if its missing - you're into prep school drilling territory.
So - tutors aren't critical, but can be useful (our one was for us, the eldest is not the sort of kid who takes parental feedback in a positive manner!). We had one for Y5 and the summer, and I think 6 months would have been fine in our case.
On Sats vs 11+ prep - I think the work done with comprehension, maths and the additional reading that a kid who is trying for the 11+ will typically do all feeds into improved Sats, so you'll get elevated Sats results in 11+ areas. Whether tutoring has an additional boosting effect, I don't know.
Finally, whilst the plural of anecdote doesn't equal data I'll add mine: I don't know of anyone using a tutor before Y5, but I do know of one family that have been practicing since Y3. Most didn't start anything until Y5, though.