Interesting, but I'm not convinced.
The raison d'etre of The Unfortunate Rake is to lament the death of a man (or woman) dying of the clap - evident throughout the genre, while the other is simply a lament for a dying lover.
The motivation for the song is not the trappings of the funeral but the cause of death.
The description of the funerary arrangements are commonplaces, which are to be found elsewhere, including the childrens' repertoire.
That the earliest version only appeared in 1790 is totally immaterial - we know virtually nothing of the oral tradition prior to the beginning of the 20th century, and even that is extremely sketchy - it never appears to have occurred to anybody to ask the singers about their songs.   
The whole 'Holy-Grail-type search for origins is beginning to worry me a little - much of it appears to be aimed at proving that the folk didn't make folk songs (not suggesting that this is the case here).
I very much doubt if there are many folk songs that can be traced back to their origins - I've never been fully convinced that any of them have.
I've been intending to open a thread on the claims that a nonsensical number of our songs originated on the broadside presses - I intend to raise this during the Ballad Conference in Limerick next week to see how far this particular malaise has spread.
Jim Carroll