Ellanwood, I am rooting for your DS. He must be soooo close to the cut off. Remember the cut off will be based on a paper application only, and there will be plenty who will be either using consultants to help prepare those applications (evidence from this board suggests it is the norm, say, for French applicants) or have strong school support. Or both.
And don't be too influenced by the "best economics student they ever had". Economics can be weird. For several years DS' academic school had very strong economics applicants rejected by Cambridge. All went on to genuinely shine at LSE. (My best guess from observation was that Cambridge interview, and lean towards the more confident banker type, whilst for economics LSE prefers studious mathematicians.) Another observation is that performance at school can be very different to performance at University. Doing well is about fit, motivation and adaptation. There will be plenty of former school high flyers who then struggle to adapt, whilst Universities have plenty of experience in judging who is right for their environment. (And obviously, given the high numbers, there is a certain amount of randomness in the process, so it will not be unusual for students who were rejected at UG level to be accepted for Masters, indeed gaining a place on the Cambridge economics Masters is often seen as easier than gaining a place on the BSc, something I assume that holds true of other degrees and other places.)
I remember from a alumni talk given by the then Director of the LSE about how careful they were to avoid over recruiting. They had miscalculated when student fees were raised and this left them with real capacity problems in what is a small and cramped University. (The talk conveniently happened whilst my son was still hanging hanging on waiting for a response. Be reassured, I stuck up my hand and explained how unacceptable it was to put 17 years olds though this level of stress. I was backed up by another alum/parent whose DC had experienced the same the year before. Though it seems that nothing has changed.)
I do hope he gets in.