Of course it's true, but the fact is that civs are represented as a simplified version of what they truly are, exacerbing clichés and making them the identity of the Civ.
You say the Mayans won't be a tall civ forever, but France is the cultural powerhouse since... well, forever I feel, while completely ignoring that France has been one of the most populous kingdom of Europe AND the most important military player of EUrope for centuries, while culturally speaking, except for managing to impose french as a lingua franca during the Enlightment, historically France has been on par with its neighbors. And yet, again and again, FRance is a cultural powerhouse without never having a military focus nor an agrarian focus.
My point is that some civs have a flavor attached to them. I know that Ireland is more than simply religious, but if we have to highlight what the Irish project most onto the world, then religion is pretty one of the major feature: one of the most catholic countries, filled with monasteries up to the brink, and basically providing EUrope with theologians and scholar for a large part of the Middle-Ages. Making Ireland not religious would be as if you made England not naval. Sure, you could, but it won't happen.
Out of genuine curiosity: if Ireland is not religious whatsoever, what could they been then?
Ireland could work in a similar way as Arabia, science through faith or faith through science, to show the big production of theologians and scriptoriums they had. And, of course, each and every new civ they could bring us could be as interesting, unique or fun as Scotland or the Gauls. But, as I said, making Ireland not religious would defeat the purpose, and we already have enough religious civ IMO, especially religious civs that cover quite large gameplays: religion+exploration/colonization (Spain), religion+conquest (Byzantium, Chandragupta, Poland), religion+science (Arabia), religion+trade (Mali, Ethiopia), religion+culture (Russia, Ethiopia, Khmers, Poland), religion+terrain (Maoris, Mali, Russia), religion+seafaring (Indonesia, Norway), religion+diplomacy/city-States (Georgia, Gandhi), religion+pop/happiness/food (India, Khmers), religion+city planning (Nubia, Japan) and my favourite: religion without religion (Kongo). Amongst all those civs, I personally fail to see which new interesting niche Ireland could fill.