Crazy crofter, the primary school may well have covered all the maths and English skills needed in the exam. However, the Grammars need to be able to distinguish between candidates and the tests will weed people out by asking them to complete an exceptional amount of work in a very short time and with a very high level of accuracy, which is difficult under time pressure. So the school might have taught the basic skills but the preparation which comes from a tutor or a parent home-prepping is about checking there are no gaps in the skills, making sure the skills can be applied to all kinds of scenarios and that they can work very quickly to time pressure and wih great accuracy. Having been taught well, being top of the top table and being on track for good SATs is all a good basis but might not be enough for these exams where there will be more children meeting the above list than there are places - so the exams have to be able to differentiate somehow and a child needs to be in that top group of candidates.
I agree that it's not great to need huge amounts of prep for many years. I agree that there will be some candidates who scrape in because of having very intensive prep and who then might struggle, but in all schools there will be those who are at the bottom and find things harder. I also agree that if all the prepping were removed from the equation and children were simply tested cold, a lot of the results would be the same. However, as were not in a system where people are tested cold, parents will continue to be concerned about their children taking that important test with the very best of chances against the many others, and it will remain that prepping seems and probably is the best way to have the best chance. Unfortunately, as some get carried away with the amounts of prep and numbers applying to some schools increase even more, reducing the amount of prep won't seem an option for most people. So saying it shouldn't be necessary etc etc may well be true, but in the current climate, prepping is likely to grow and not reduce.
Schools which use VR and NVR to test also add a skill not taught in state schools - a weird and unusual kind of test which many parents won't feel confident about, hence the need to certainly familiarise and often by tutors who have experience of this.
Independent school exams might be similar and different. In London, there is a lot if competition for some schools, but with many sitting 5 or 6 exams, the applicant numbers distort the level of difficulty of getting in as each child can only attend one school. Even with the very competitive nature if it all (and it might seem more competitive due to obsessive parental chat about it) I'd say children generally don't need to prep as much as for the super selective state Grammars and can be and often are a bit more relaxed. First of all, by paying, people have lots of choice. They aren't restricted by catchments and there are lots of independents where there are few superselective grammars without catchment, meaning parents can apply to lots if they wish and if they apply to a range of competitiveness, are sure to get somewhere in the end. Applications which would put the school as first choice,per place are also crucially lower - due to fees reducing the numbers who can apply - and this of course makes it easier to get places. At all but the very very top tier, and often even at those, children get offers, and even scholarships who didn't pass the superselective state grammar tests - in the end its demand and supply. So although many independent parents also prep and pay tutors and get in a stew about it too, the level of preparation and obsessive focus on prep I think is lower. Being able to pay simply gives you more options which relieves some pressure, which those with the only alternative being a terrible Comp (and of course many have a good Comp as the alternative, but still don't want that) don't have, and so are more driven.