"Every other part of the world outside Ireland, regardless of language, has no problem with the labels: Ireland and Northern Ireland."
That's just not true. Which other countries have you lived in?
Between DP and I, we have lived in 7 different countries across 3 continents. I'm not going to list them because it's an unusual mix of places and could be identifying. Why? Where have you lived?
My point doesn't come (solely) from personal experience. What clinches it for me is my friend who works in the post office in Ireland, for whom this is a pet topic. Post arrives from all over the world with "Ireland" as the country at the bottom. A very small number come marked "Republic of Ireland", "RoI", or similar, and these items almost all come from the UK (based on the return address). If the country's name is also given in another language (Irlanda, Ирландия, etc.), it's invariably of the form "Ireland" and not a "Republic of..." version. In other words, people in other countries are happy to use Ireland as the label for Ireland, but there are a few folk primarily in the UK who use RoI.
As for Northern Ireland, my friend in the Irish post office encounters a small but regular volume of mail that is labelled "Ireland" but has a NI address and postcode. These items all (to her recollection) originate in England, Scotland, or Wales and hence don't even need a country name (i.e., the postcode would be enough) but if they have a country label it should be "Northern Ireland". The Irish PO redirects them to NI. There is also a small volume of mail that is correctly labelled "Ireland" and is destined for an address in Ireland rather than NI, but ends up in Belfast. Again, these items pretty much all originate in England, Scotland, or Wales, and have been mis-sorted into international rather than domestic mail. The NI PO redirects them. (You'd think the domestic vs. international postage would help in these cases, but apparently there are errors there too). In other words, although it doesn't happen often, there is sometimes confusion between the labels of "Ireland" and "Northern Ireland" when both addressing and sorting mail, and when it happens it overwhelmingly comes from Britain rather than any other countries.
So unless someone has better evidence, I'm going to stand by my assertion that every other part of the world has no problem with the labels of Ireland and Northern Ireland.