Shootingatpigeons, my DS scored 85th (VR - 1 in 7) and 99th (NVR - 1 in 100) percentile on school testing (done for assessment purposes, not admissions prep).
After some prep, in which both scores improved, he came out around the 50th percentile within the population of children sitting the test for both subjects when the results came from Tiffin.
So I would say it's nonsense that they are looking at the 97th percentile.
He got into LEH's brother school (Hampton), and they do do these tests as well, so I think you are right that the vast majority of successful girls at LEH WILL score beyond the 95th percentile, but that is 95th percentile based on a cohort consisting of the whole population of 10 year olds PRIOR to doing prep, which certainly all prep schools would prep for prior to sending any girl along to testing at LEH or other school conducting such a test.
So it is deeply misleading to compare Tiffin, where the top candidates will score in the top 0.001% or whatever of a population of random completely uncoached children, but only perhaps in the top 10% of the population of those sitting the test with a bland statement about the scores that LEH gets.
Private schools do not generally have better versions of the tests than Tiffin does. What they do have is the freedom to interpret them how they wish, i.e. the kids who do poorly get disregarded entirely, and those at the top are identified.
What Tiffin and others do, which is absurd, is to disregard the fact that the tests aren't really capable of discriminating beyond around the 99th percentile, yet they get 500 kids all scoring 97th percentile or above based on national standards, and they try and rebase those 500, every one of whom deserves a grammar school education, based on a almost random stack rank (ok it's not random in that it's based on which child did best, but the tests WEREN'T designed to discriminate between so many high performing kids), within the essentially arbitrary context of the kids sitting the test.
The grammar school will say a kid on the 97.6th percentile for some set of silly tests is better than the one on the 97.5th percentile, and reject the second one. The private school will look at reports, interview the child, and say 'actually the second child is really gifted, and the first one has prepped to within an inch of his life'