They have various nationality sections, but have weirdly subsumed the few Irish tunes I could find under 'British'. Do they not realise what offence this will cause?

I couldn't find many, but the first Irish-connected song I could find was "The dear little shamrock", sung by two Welshmen. (I now know where the tune for "M-hm" came from; thanks).

Trying to avoid causing offence to Irish-Americans is a waste of time.