In the case of "Callino casturame", there's no doubt where it was first written down and there hasn't been any dispute that it's Irish for a couple of centuries. So you're attacking a strawman.

It sometimes gets more interesting. The earliest copy of "Deil amang the Tailors" I've seen was an 18th century Scottish manuscript that titled it "American Reel". It's always been a very popular tune in America but the Americans all think it's Scottish.