Anyway, to answer OP's actual original question, I don't believe any child could get into a grammar school with tutoring, let alone a super selective one.
In state secondary schools, I've met Y7s who struggle to read and do basic maths. I've met Y11s who can't access a GCSE foundation paper for my subject. I've also met students who panic under exam conditions, no matter what prep or support they are given. I've met students who needed heavy parental support, or additional tutoring to scrape 4s in English and maths. I've met hard working, lovely students who struggle to finish exam papers but don't appear to have diagnosable SEN.
I don't know that much about 11+ exams (beyond sitting them myself), but from what I can remember, I would think at least 25% of Y7 in an average state school would struggle to access the exams, even if given tutoring- this isn't including those with significant diagnosed learning needs. I think the next 25% would probably struggle to achieve the pass mark even with intensive tutoring and all the support in the world.
I would think that for the most selective London grammars, you'd probably need to be in the top 50-25% of the cohort to stand a chance- and probably the top 25% if that tutoring wasn't very intensive.
Equally, I think students who might be in the top 5-10% on raw intelligence may struggle to get into the most competitive schools if they haven't been prepared by someone. I'm sure there are students who get through with no or minimal tutoring every year, but they must be a tiny minority.
My experience is also that a lot of students at grammar and private schools do get tutoring at various points throughout their school career.
Locally to me, which is pretty far away from London, there are a small number of grammars in the next LA- they might be considered super selective, I think. We pick up students in our sixth form who've gone to the grammar schools every year. Some of them are amazingly bright, some of them are B/C students if they're not being pushed- but no lower than that. And we have students who are D/E students, and equally those who don't get into our sixth form or don't get accepted onto A-levels. Equally, I've got A* students who've gone all the way through a comprehensive system.
I do think, to some extent, grammars are also selecting for supportive families and active, involved parents.