Although I have been reading Middle English for forty years, on and off, including the Gawain poet - and more recently Langland's Piers Plowman - and although I knew in theory how the alliteration was constructed - until today (this very evening) I never understood how the alliteration did the primary work of poetry which is to make it memorable.
Or rather, memorizable - by the poet.
...That being the primary origin of poetry.
(And the un-memorability of high status modern poetry (coughSeamusHeaneycough) is a clue to the fact that it is not real poetry - not the same kind of thing which poetry was in pre-modern times.)
Anyway, today, this evening, the penny dropped. I worked-out for myself - piecing together clues from re-reading the descriptions of the mechanisms of alliterative verse especially by JRR Tolkien and CS Lewis - and adding the principle that memorability was the key.
Thus I worked out 'in a flash' how alliterative poetry must have been written, remembered and recited - and then I was immediately able to confirm this by finding some recently-posted (and not previously viewed) informal YouTube recitations from a young expert in this topic at Leeds University - one Dr Alaric Hall (whose PhD thesis was on elves! http://www.alarichall.org.uk/phd.php) - which completely confirmed my thesis.
The basic structure of Old and Middle English alliterative poetry was that it was written in two half lines - each half line with two stresses which alliterate twice in the first half line, and then on the first stress of the second half line.
Alliterate - Alliterate // Alliterate - Not
The key insight was from Lewis's essay (in Selected Literary Essays) when he said that the three alliterating words in a line carried the essential meaning of the line.
This suggests that the poet constructed the poem around the three alliterating words in each line, and remembered it by learning these three words, and recited it by stressing these three words.
It is likely - given the way that memory works - that the unstressed words and the non-alliterating stressed word in each line were skipped-over, somewhat variable, and probably paraphrased in actual performance-from-memory.
How would this work in recitation?
Well not in any of the ways that I had heard alliterative verse performed by actors - who focused on the meaning of the lines, as if they were prose.
But it would, I expect, have sounded like this:
or perhaps even more like this (the first part of the same passage, done in an even more un-inhibited style):