Amos that is surely an Irish version of the song, couldnt agree more. But that is not evidence of Irish immigrants having taken that song to the wild and wooly West and using the tune for "Git along little dogies". Maybe they did, maybe they didnt. The fact that you've heard an Irish version of the song doesnt throw any light on the question one way or the other. That's the only point I'm making.
Slainte/cheers