I think of such movements as rudimentary dancing: the singer is feeling the rhythm but cannot stray from the microphone, and is therefore limited to moving arms and legs. If you can bring yourself to watch contemporary pop stars, you'll find most wear headset mics - for the very reason that they enable the singers to dance.

Another aspect of hand movements is emoting - emphasizing what is being told in the lyrics. Since Karen Matheson was mentioned, this perfectly illustrates how the gestures help tell the story. (I defy anyone to get through it with eyes dry.)

If a singer doesn't have an instrument in his hands, what should s/he do with them? Sticking them into trouser pockets would surely look odd to someone... as would hanging them straight down, folding them in prayer, balling them into fists, or anything else one could conceive of.

PS: How did I know that Jack Campin would jump at the chance to slag off the inimitable Eddi Reader?