No idea how it is in Ireland, but in most of the EU, digital transfer between banks and countries is completely seamless. I can transfer money to family members accounts in different banks in the same country in a couple of minutes, and to another country in a day or two. I haven´t used a physical cheque since I moved back from the US.
I´m sure American banks have very competent teams. So do the Europeans. Behind the scenes they all also tend to collaborate quite a lot in this area at least.
I'm with Ulster Bank ROI (which is actually a subsidiary of Royal Bank of Scotland) and have had a fair share of minor issues with them. Their app also sucks big time.
They've had a major issue 6 months ago where a fair amount of folks were locked out of their accounts (or couldn't access the money, same same ultimately).
Friends at AIB have also had issues, like discrepancy between your online statement and the paper one, or getting their account straight up closed out of the blue.
I can't remember any specific with BOI but they also suck.
If you're in Ireland for long enough, do yourself a favour, get a Revolut / N26 account for your day to day, it's pretty smooth.