The idea that a priest wouldn't hump a harp around is predicated on the harp being the modern concert harp - that would be unlikely all right - rather than the small metal-stringed harp used by earlier Irish musicians. Like Carolan, Bishop Donnelly is said to have been an itinerant musician; the harp was a good disguise for a priest, since only two bishops were left alive in Ireland at the time by the raging anti-'Papist' English who were determined to wipe out both Catholicism and Irishness.
The tune is typically Irish - typically northern, even; the words are typically 19th-century, but also, I think, typical of the softening and simplifying of more complex and intellectual verse of the previous century in Irish.
The fact that the story comes to us from béaloideas rather than being written down by scholars does not make it untrue; how many stories of today's South Sudan massacres will be lost if only the ones typed up by journalists or in UN reports are later believed?