Steve,
The fact is that neither of us have one iota of proof on whether the songs originated on the broadsides or in the minds and mouths of 'the people' - it is a matter of opinion - and because of this I believe it would be presumptuous of either of us to make definitive statements without providing evidence to back up our claims.
Since you first raised the point some time ago I re-visited some of our books on the broadside trade (the Catnach and Pitts bios., Roxborough and Bagford collections... Shepherd, Collinson et al,). There is no proof one way or the other - or if there is, could you point it out. It seems to be totally a matter of speculation. I would not deny for one minute the possibility that some of our our songs and ballads MAY have started life on the broadside presses, but without adding to my present knowledge it would be extremely arrogant of me to say how many or which.
If you are prepared to concede that the bothie songs came from the Aberdeenshire farmworkers, why are you not prepared to accept that the sea songs came from working sailors, that rural songs came from rural dwellers......... what makes the bothie workers so special?
I have given you an example of a rural community in West Clare which not only had a large repertoire of traditional songs, but also had a tradition of making songs using the old forms. This was also the case with the Travellers - they continued to make songs right up to the point where their singing tradition died (some of them as short as 2/3 verse pieces, others extending into 7-8-9 verses).
In 1975 we started recording a Kerry Traveller named Mikeen McCarthy. In the 1940s he and his mother participated in what was the dying gasp of the broadside trade here in Ireland - ballad selling around the fairs and markets in the rural south west. He described to us in detail how those 'ballads' were selected and produced.
At the risk of making this an epic of this posting, this is how one of the many discussions we had with him went.

J C.       Where did you sell mainly, where did you sell your songs?
M Mc.    Fair days now, inside the pubs.
J C.       In Kerry, or would you travel out as well?
M Mc. Oh, I'd travel away too, Kerry, Clare, all over, wherever there'd be fairs, anywhere you'd go when the fellers'd be half steamed in the pubs, 'tis then they'd start buying them.   
J C.      You say your mother would sell them as well?
M Mc   Well she'd never hardly sell the songs that she wouldn't know, because she couldn't read, you see. But she'd sell the songs she used to know.
J C    How would she get them written out, would she get somebody who could write to do it?
M Mc   Yeah, the printing office we used to go to now, he knew us that well he'd have them all ready wrote out, so she'd want a gross of those songs, that's twelve dozen, twelve dozen of the next songs he knew her well like; "now Jane, I've The Wild Colonial Boy", for instance or "The Blind Beggar", we'll say, all those songs, "I've all those in print now".   They'd all be laid out on the counter then in all different colours, there'd be kind of pink, orange colour, yellow, and white, all that, you know, and they'd be all in bundles like. Well you'd pick and chose them, whichever one you want, about threepence a dozen I think that time, fourpence more times.
J C    How many would you sell of each song, what would be a good sale?
M Mc   Well, 'twould be a long day's selling like, and if it would be a big fair, if I sold say two or three dozen of each song, you should sell at least a gross anyway, like, twelve dozen.
You'd go into a pub, only you'd have the ballads in your hand, just walk over to the group and you'd say, "would you like to buy some songs, some ballads".    They'd start looking at them then.    Well they'd take them all away, they'd start reading them all then and picking them out and they'd ask you then, "could you sing that one for us, could we know the air of it".    "Yeah", I'd say; I'd sing it then. They'd buy me a bottle of lemonade or something and I'd sit down and I'd sing it and then I often had to sing it maybe two or three times 'cause there'd be some girl maybe or some boy interested in it.    Then they'd want to get into the air of it like.
J C.         So you did in fact teach them the air.
M Mc.   Yeah, you'd have to teach them the air and they'd have to go over the ballad then again and maybe I'd have to sing it again with them, you know, but they wouldn't want your time for nothing, oh, they'd pay you very well, whatever you'd want to eat, or something like that, inside in the pub.   'Tis like the records now, it reminds me of the same thing Jim. You'd get a hit ballad, so I'd get that in print straight away then. But 'twould just travel through our parish or through a town, from one town to another, and fair to another and you'd get the new ballad come out and you'd sell twenty times as much of that ballad as you would of the rest of them, when they come out new like.   The Blind Beggar sold very well now, that one. All those songs now, The Wild Colonian Boy; several songs like that now.
J C.    What would you say was the oldest song that went onto a ballad that you know?
M Mc.   Oh, The Blind Beggar, I'd say, I'd say that was the oldest.

He selected song from everywhere, but one of the main sources was from his father's repertoire of traditional songs, sometimes by request,

"Do you have any of your father's songs? -
"Not today, but I'll have some next time".

None of this, of course, is solid proof, but it is an indication of how the songs were transmited.
Incidentally, I believe a couple of The Robin Hood Ballads did pass into the tradition.
Regarding your Arthur Howard observations(would be interested to hear how you approached the subject).
With both the Travellers and here in West Clare, we were working in a situation were the singing tradition was either still a living one or only recently dead.
The situation in the UK was very different, traditional singing having been swamped by the popular songs of the day some time in the early part of the 20th century.
Walter Pardon described how he was the only one of his age group, in his family, to take an interest in the old songs - the others having opted for the new ones. Walter had some fascinatoing things to say on how he differentiated, as I'm sure you know.
Mike:
sorry - a slip of the mind (i or i) should have read (d or i) - directly or indirectly; from traditional singers recordings or via books.
Jim Carroll