There are loads of stories about quick learners. Busoni apparently once met a former pupil on a train from London to Brighton (even in those days barely over an hour's journey). The pupil showed Busoni his latest composition, which Busoni handed back before the train arrived. At the concert that evening, the pupil was rather surprised to hear his piece played as an encore.
I've heard the story about John Ogdon and the Brahms concerto but can't remember the details. However, here's one I witnessed: at the recording of Sorabji's 'Opus Clavicembalisticum', after John had played one particular passage, Alistair H pointed out that the score was incorrect (it's full of misprints, not surprisingly) and in fact, of the three staves on one system, one was in the wrong clef. This of course meant reconfiguring the passage completely, which many people would agree is worse than sightreading. John nailed it first time. I also suspect that John sight-read Grainger's transcription of the Tchaikovsky 'Flower Waltz' in a concert in London. The piece includes a misprint so obvious that on the first play-through any pianist would stop, pencil in the obvious correction, and never play the misprint again. John played the misprint.... Ronald Stevenson once gave John a newly-composed piece to read and was turning pages as John did it. John was nodding for page-turns a whole system before the end of the page, and when RS took this on trust he found John had indeed memorised several bars ahead.
A friend told me a story about the pianist Ronan Magill. My friend had asked RM if he knew any piano music by Woldemar Bargiel (no, I hadn't heard of him either). Magill replied that he didn't, but said he had turned the pages for a performance of the Bargiel violin sonata about 20 years before. He then played a few minutes of it on a handy piano, incorporating the violin part.
As an opera repetiteur, I earn my living (partly) sight-reading and I'm fairly handy at it. Some of my colleagues are pretty impressive. JP Gandy allegedly once sight-read the orchestral part of Prokofiev's 2nd concerto from a full score. Helen Crayford plays 'plinky-plonk' contemporary opera at sight from full score. I've heard Jeremy Limb (also a soloist) sight-read the odd piece and he's one of those whose sight-reading is not just note-spinning, it actually sounds like a performance. A few years ago, a retired singer gave me a tall full score, transposing instruments and all, and asked me to play it a minor third down: she seemed genuinely surprised when I had difficulty, so I guess she must have met pianists who could do it.
These are all aspects of quick learning. None of them, however, guarantees that the final result will be a moving musical experience!