I wonder whether exam boards don't set really well-known pieces, with a long history of multiple famous recordings
Except, of course, that they do set well known pieces with a long history of multiple famous recordings. Finzi's bagatelles for clarinet have seldom left the clarinet syllabus, movements from Haydn's Trumpet concerto constantly appear in the trumpet syllabus - Brahms' Lullaby is there in the piano syllabus, and Chopin's Minute Waltz has been set for Grade 8, Fauré's Sicilienne is a favourite choice for flautists, and so on. Just because Claire de Lune seems to have only once been set for a piano grade should not, I think, be taken as a precedent for assuming that well-known pieces are not set.
The difference is that the repertoire for piano is so vast compared to other instruments. So particularly for wind and brass high grades, the same classics tend to keep coming round as syllabuses are refreshed, while each new piano syllabus brings many pieces that haven't been featured for a long time, or which have never appeared before. It's a very general trend and individual pieces buck the trend, so new syllabuses on other instruments include new discoveries, just not as many as for piano. And on piano, the most over familiar pieces do appear from time to time, just not as often.