Why not Do thábhairne?
Because tábhairne by itself doesn't mean 'pub(lic house)"
Then why is it given as a dictionary hint in the first position - or at all?
Most likely because of how Duolingo handles words. If you see an incorrect dictionary hint, please report it.
And in America, the words are both rare, and both synonymous with "bar" so the difference is lost here.
Another person said "tavern" is basically never used in Ireland. Is tábhairne used by itself in Irish to refer to something different?
I've never seen it by itself, but it supposedly means 'tavern' by itself.
Which doesn't seem a distinct concept from "pub" in American English, except perhaps in licensing or historically. Someone else claimed "tavern" isn't a term much used in Ireland. Do you know if that's true?
tavern isn't used in Ireland, as far as I know.
It never occured to me that pub was short for anything