I think as oboebunny says it's simply a matter of technical ability and time constraints. (Although we could ask why Classic FM only play a recording of one movement of a concerto when leaving the CD on isn't hard but that's a different question

). It's not only the technical difficulty of the other movements being harder than the chosen movement but the fact that learning a whole long piece is quite an undertaking in itself.
I seem to recall that for some woodwind instruments (I use this example as it doesn't happen so much in piano music as the repertoire is larger) certain sonatas/concerti appear with, for example, one movement on the grade 5 syllabus, one on the grade 7 syllabus and one on the grade 8 syllabus, with the entire sonata being on the dipABRSM or LRSM syllabus. One could argue that a grade 8 student would be able to play the whole piece as it's all within their technical grasp; but the stamina required to play the whole piece, and the ability to tackle long works are skills in themselves. It can be quite a daunting undertaking to tackle a long piece, knowing where to start is important. Of course a performance at diploma level will be expected to be better than one at grade levels as well. Furthermore, if a diploma level player has to tackle a grade 5 level movement it will consume less of their time than it will for a grade 6 level player: thus more can be taken for granted and more ground can be covered in the same time (not that it will take no work on expression etc.): it's not trivial that a grade 8 player can musically play grade 6 pieces instantly for example.
It's actually quite usual to find the middle movements at the grade levels (in piano sonatas, I don't know about other instruments) since, as Hannah points out, these are
usually the technically easier movements requiring expression more than technical wizardry.