You're be talking about high CAT scores for the top super selective Kent Grammars too. Remember the huge numbers who apply because the schools are free, which make them hugely competitive. On an applications per place basis, they will be more competitive than the independents which simply have less applicants because of fees.
Yes, consider SevenOaks - you might get a scholarship, but it's worth remembering that most scholarships are not worth much money now and unless you qualify for a means tested bursary, you will have to pay most of the fees, unlike the Grammars.
I actually think that most of these selective independents take a wider range of ability (measured by CATs or other methods) than people sometimes think. It is definitely in a schools interest to push the idea that very high CATs are required and I'm sure that high levels are, but probably not at the levels mentioned. These schools have more applicants than places, but the kids who apply also apply to a number of schools and can only take up one place. We all know that schools make more offers than places because not everyone will come - those who get offers might range from the top 141 score for a number of students, down to a fairly significantly lower score at the bottom of the offers list, simply because they need to fill their places and only have a certain number of applicants who can afford to pay the fees.
If you speak to the teachers in these schools, you will hear that yes, the students are really bright....but there is still a range and some are not as clever as others.....but they got in. Every school has those who struggle a bit to keep up with the high expectations - who are at risk of a B or C at GCSE or A Level, rather than the A*. I know that these grades don't correlate with CATs but the point is that although all abilities might be high, they are not all super-high as is suggested by some people.
It is good to be guided by Preps are CAT indications are useful. However, I would say that with scores of 120+ its possible to have a go anywhere. And Inwouldnt be at all surprised if those top schools have a handful of students who are actually more like the 110-120 CAT mark too. Great schools can get all A/A* out of students with 105-115 as well those with the higher scores. And I know it's not all about exam results and high CATS might indicate abilities not measured by GCSE or A Level, but schools are still very very interested in these grades.