I'm sorry if my post was surprisingly unhelpful.
I only wanted to point out that the original poster (in her first post) listed only a handful of schools that included the most well known boarding schools in the UK.
I do know some things about the boarding school admissions process so I'm not wholly ignorant.
At schools like Eton, in any given year there are only a certain number of slots available.
Eton has to consider all the traditional prep schools that feed into Eton.
Eton has to consider all the siblings.
Eton will want some diversity, it isn't the purely establishment school it was a few decades ago and they do want some international students, some students from interesting backgrounds as well as students from families that have a strong historical link with Eton.
Eton has a bunch of other priorities in its admissions process that only you and I can guess at.
Very bright boys will be rejected from Eton and it's because there isn't space. The number of spaces available in any given year to a boy who doesn't have one of the above advantages is extremely limited. The original poster is coming late into the game compared to many families who have had Eton and other schools of similar calibre in their sight since their children were born and their names on waiting lists as well as preferred feeder prep schools, and that has put her son at an additional disadvantage.
If you're only applying to Eton, Winchester, Westminster, Tonbridge and Harrow and so forth, especially without coming from one of the traditional feeder prep schools or family connections (the father or grandfather was a student) you are facing stiff odds. Not impossible, but stiff, although each school will have slightly varying odds depending on the nature of the applicant, Winchester, for example, loves brains and will look more closely at genuinely bright boys, Tonbridge likes sporty pupils and so forth.
And the odds do vary from year to year, once the preferred applicants from the prep schools or those with other sibling/family/diversity connections are admitted, the number of spaces available can be just a handful or can be many more. It might only be six seats, it may be 20! But since we can't guarantee what the chances will be as it's never a neat formula, it's always wise to look at a wide range of schools. I'm sure her child is extremely capable and a good school will certainly take him. But then we get to the other half of the story, finding the best school once you've been admitted. You want a number of schools to compare before making your final selection, it's no fun applying to only five schools, rejected by four, and ended up having to go to the school that was the least suitable because it's sporty and your child isn't sporty.
I get the impression she's not very familiar with the larger boarding school scene in the UK and there are many more just as excellent schools that are easier to get into and that may not have been on her radar. So that's why I nominated a bunch more very good schools for her to consider.