Hmm. Now you're asking. If you want my honest opinion I'd say no in this case. Also we're talking about a different era. Teddy O'Gra in its original form of the early 19th century, definitely stage Irish skit. I don't see this in Joe Geoghegan's song of the late 60s. Geoghegan presumably was of Irish extraction though probably born in Bolton, Lancashire. His songs are about 50/50 serious/comic.

Nobody has ever suggested any 'over-sensitivity of the Irish towards their war-wounded', quite the opposite in fact. I'd say the O'Gra song is pitilessly insensitive and nasty. People still sing it as an anti-war song though and why not? There are lots of songs in the folk repertoire that are now sung as serious songs that started out as comic songs.