I mean, I don't expect this to be enforced for most travellers but I suspect if you want to stay in NI then you'll probably need to show a visa (based on current UK law).
I rephrase, technically I haven't needed to carry my driving license for the last several years of my driving career, but legally, I may be asked to present it at any time. I'd take the advice of the website cited above rather than believe you. I.e. you could be asked to prove that you're a UK/Irish citizen (easiest with ID or passport), and if you're a EU citizen entering the UK (the website even mentions crossing the Irish sea to enter Great Britain), you have to have a passport (although you might not be asked for it).
Before Brexit, the UK was part of the EU, and other EUians was allowed to enter it with just their ID card. So...