My question is about the history of the Passover song Chad Gadya. I have long suspected this song was originally conceived as a pastiche of Echad Mi Yodeya--the thematic, musical, and phonetic similarities seem to me too many to overlook--with the Aramaic lyrics pieced together around a central theme over time and across Seder tables. However, it struck me that there is an even more obvious correlate of the title and refrain of "Chad Gadya": the "Haggadah" itself, or at any rate its title.
This is admittedly a long-tail question about something it may or may not be possible for musicologists to answer, if they haven't already; still, I will give it a try. Are the phonetic similarities between "Chad Gadya," "Echad Mi Yodeya" and "Haggadah" a matter of pure coincidence, or can anyone provide historical evidence of their intertextuality? Put another way, was the original refrain of "Chad Gadya" "Haggadah"?