Dispute greatly your original statement of
"there is no shame in not getting in"
Maybe shame is the wrong word. It's a stonking huge kick in the teeth getting a rejection from any of your uni application places, and Oxbridge more so it seems ( as you will have been pre selected, pretested and got through all that, to be rejected after interview). Up until the past year or so the rejections from oxford came handily on 23/12 to ruin Xmas!
Yeah to improving resilience etc but really it's a huge thing to apply, to think you might be good enough and then to be told you aren't. My oxbridge applicant and mates in the following year groups are helped by the fact it is a school that sends to Oxbridge often ish. Therefore there are many " rejects" who support each other!
Not something I'll encourage the youngest to do.
THe "reject" is top of the year at a russell group uni with a funded masters lined up for sept, and a paper accepted for publication. So yup maybe not right for Oxbridge but certainly not "substandard" as they felt at the time.