To sum up: Irish people are not offended.
Wrong. Rtft properly. Opinions amongst Irish posters are mixed. There are plenty of posters from Ireland (the republic bit, since you want the make the distinction) who have said repeatedly that it's a shitty offensive phrase. Stop ignoring us.
Last time a thread like this came up, there was a big divide between Irish posters who had versus had not spent a few years living and working in Britain (as in the island if Britain; NI is a different kettle of fish).
Those of us who had lived for a while in Britain were more likely to object to phrases like this because we were contextualising them against a constant undercurrent of anti-Irish sentiments we had personally encountered in Britain, compared to Irish posters who had mostly lived in Ireland or whose time living abroad had been in Germany, France, Middle East, etc. where those undercurrents were not so present.
I don't know if the same dynamic is happening here again. However, after you've repeatedly experienced and observed anti-Irish abuse and denigration in Britain (in recent years... this is not about times or generations past), the scales somewhat fall from your eyes, and it is not a harmless turn of phrase when British people casually use language like Irish twins, throwing a paddy, a bit Irish, etc.
Othering feeds discrimination. Every time someone uses a phrase like that without thinking (or without caring; whichever) it helps to normalise a culture where someone else feels it's acceptable to call an Irish person a stupid fucking Paddy, or to laugh at an Irish university degree as being worthless, or to say that someone is pretty smart for a Mick.
Just don't.