Don Firth's got it. Rhyme points us to the mnemonic dimension of what is sung. It also had the quality of what Walter Ong called the "paideutic spell" -- a hypnogogic tendency that enhanced learning, sent whatever was heard not only to memory, but to whatever part of the mind is opened when we're in a suggestible state.

Rhyme is only one kind of sound-patterning; it becomes the norm in English after the Norman conquest. Traditional Germanic poetry was alliterative, but tellingly, they called it "Stabreim," that is, the words rhymed at the beginning, not at the end of the words. Then there's assonance and consonance, syllabic quantity, all kinds of other features.

Best,

Adam