"One thing that IS true: ballads as such are far more characteristic of English and Scots culture than of Gaelic culture, so the
ballad tradition is largely an import to Ireland. But Irish folks sure ran with it! "

Oh and what makes you so positive about that?

Second if true - doubtful - it helps not your claim, since you are talking about lyrics while the debate is about tunes.

The English and the Scots do not share the same musical culture to the exclusion of the old home, Ireland, in fact quite the opposite.

In the case of the Rocking Cradle theme it is more likely not British as one poster already suggested.

Merely writing a Book about tunes and claiming them for the county where collected is not, nor could not, be evidence for anything other than some speculative publisher wanted pages with something however unlikely.

AND English language songs often found new melodies, even Irish ones!

Gospel Music is full of this kind of thing, there are thousands of ripped off tunes in these books but who bothers to document that? If they included a modern Chinese pop theme who would notice or care?