You're making a straw man argument there. I've not said that, and nor do I think it; I was responding to and quoting from a suggestion in the Sutton report, comparing the bottom end of kids qualifying for grammar school admission - the tail end - with the small number of children from very seriously deprived backgrounds who would be likely qualify via threshold, but who would otherwise not be so far below the more privileged. Those are the only ones who would be affected, and therefore my statement is perfectly accurate. Deprived and uncoached children achieving only slightly below very privileged and heavily coached ones have done better, when relative starting points are factored in (provided of course you believe the tests to be valid in assessment). Perhaps you might want to read what people say before, what was it - making lazy generalisations and massive assumptions? 
I'm actually privately educated: plenty of kids at my school were extremely bright, and really flew with the additional advantages such an education afforded them. Same thing at university, where a large number of students came from private schools (and there was a very active conversation about factoring in overall attainment at their schools of origin to assess quality of support in gaining the necessary A level grades - there was an understanding that the system was not fair as it stood, because a private education doesn't surgically remove conscience). I really don't need anyone to tell me that kids at private school are as varied and able as any others - just a damn sight more privileged. I'm fully aware. And I have never suggested that kids from private schools should be prevented from attending grammar schools, either. Others on this thread have, but not me. My points have solely related to addressing how very teachable the tests currently are which entrenches yet further the privilege of the privately educated/coached; that I think suggesting a threshold entry for kids from the very poorest backgrounds, rather than the ranked system applied to everyone else, is a good idea; and that arguing that privately educated kids would miss out educationally if denied grammar school places is fallacious. They wouldn't. That doesn't mean to say I think it would be fair, or should be done. Plenty of children at good state schools are only there because their parents can afford stratospheric house prices in that catchment and then sit happily on capital gains while benefitting from an excellent education, while some who can't afford to buy their way into a top comp, and faced with a sink school, send their kids privately by taking second jobs - each - borrowing, and scrimping every penny they can. Others are sent privately because they're horribly bullied and there isn't another option within the local state system. I don't think a state/private cutoff works, personally. But nor do I think it's true to say the children will be deprived - they won't, their parents will. Their parents have already demonstrated a fierce determination to ensure their kids will have the very best education they can possibly provide. It's a disingenuous way to appeal to sentiment, saying "think of the children!" when their parents are already doing that very thoroughly indeed. In reality, two groups are competing for additional benefit at state expense, and it's perfectly legitimate to argue as to how best to award that benefit.
Back to the main point: if the small number of kids from very poor backgrounds (admittedly the extent and persistence of that poverty is always a guess anyway, because some confounding factors might well be present - parental relationship breakdown suddenly affecting family income, or unexpected redundancy) are offered places ahead of kids from private schools, then the only kids to miss out will be those on the tail-end of the admitted places, because you'd need a fairly high qualifying threshold for the educational opportunities to be ones everyone could benefit from reasonably equally, so bluntly very few of the very poorest would gain admission even under those new rules, anyway. The nature of the ranked admissions system should mean that those kids at the very bottom end of the admissions cut-off are likely to be the intensively coached but average children (of course, that's always assuming you think the current tests are valid as an assessment of potential/intelligence, which I actually very much do not, but let's leave that for now) which means, yes, that a child from a deprived background has done better to reach that stage unassisted, and would probably benefit more from the place.
Honestly, I don't think anyone is listening to anyone else on this thread. It's a bunch of entrenched positions and easy targets. I think the exams should be adjusted so they're harder to coach for - nothing is uncoachable, but by the same token expecting coaching agencies to talk down what they could contribute under an amended system is turkeys and Christmas and positive changes could be made - and would allow the very bright to shine irrespective of coaching. I think the very poorest should have a lower threshold, but it would still need to be a very meaningful one to ensure the child in question didn't struggle. And there could be more done to seek out innate talent and weed out coaching, which actually is a bit unfair on the less able who can't keep up, as well as the more able who don't get in. That's all I've actually said.
I think there needs to be a discussion on how you level out educational opportunity in this country, and I think people should leave their pet prejudices aside if it's to be constructive - but none of us are capable of that. It's just human nature. So I think at least we could seek to tinker at the edges, as suggested above.
Academic achievement isn't just about natural talent though. Natural talent can only do so much to compensate for a disrupted home life. If a child does have a disrupted home life, which can in no way be assumed just because of the pupil premium, then they are likely to struggle at grammar school anyway.
It takes motivation and the opportunity to focus as well as natural talent to do well in a very academic environment.
It isn't about a "disrupted home life" if you read the report I linked to - the assumption you refer to there isn't mine but your own. There's a straightforward, statistical link with poverty itself, even after you account for other factors. The pupil premium very much does track that. If you're seriously arguing that kids without coaching and from poor families, who are statistically likely to do far worse academically, haven't demonstrated their suitability for academic study as well as their "motivation and ability to focus" by passing a rigorous entrance test, without coaching, when almost every child they compete against has been intensively and expensively coached, then I'm afraid I don't think we have much more of interest to say to one another.