Again though, (very) selective private schools, unlike grammars, are not looking simply at those who achieve the highest mark. So, as with any exam, you need to check that he has covered the material and make sure he has, and give him enough practice that he gets the timing right, but too much more than that can be counter productive.
Schools are interested as much in what kids can contribute to the school as much as what they offer the child. So some will gain places because of their sheer academic horse-power. But there will also be places for children who are good enough, but who also contribute to school plays, choirs, orchestras, sports teams, leadership etc. Schools will see education as something much wider than exam grades, and the schools you are talking about really don't need to buy in grades. The same will be true, perhaps more so, with bursary candidates, who hopefully contribute social or economic diversity as well as engagement in the wider life of the school.
A statement on the lines of "he really loves sport and is good at it so I wondered if he may sway toward sports at the expense of his academic potential" might cause concern. There is a noticeable "my child first" minority in some schools, which I suspect schools find frustrating. The founding principles of some of the schools mentioned are often more about education bringing responsibility. I suspect that at some point a child gains less from tutoring and more from spending their time reading, playing sport, hanging out with friends or doing things with their family. Academic intelligence is one thing, but emotional intelligence is what will see a child through the educational long haul.