On this, we'll have to agree to disagree. Culturally, the Republic of Ireland is certainly to the right of the United Kingdom (for example, abortion is flat out illegal in the nation of Ireland, whereas its perfectly acceptable in the UK, and even the so-called "Conservative" Party leaders won't restrict it). Part of it is a cultural divide and religious influence is certainly a factor as well -- the Catholic Church as a whole is well to the right of the sleazy Church of England, the "official" religion of the UK.
The fact "Catholic" Sinn Fein is a bunch of marxist atheists/closet terrorists (who I am delighted to see lose, BTW) and the DUP is a decent, reliable conservative party (certainly to the right of what passes for "Conservative" in the UK) won't change the external factors being the opposite. It reminds me a bit of the post-Civil War era south voting for the likes of Woodrow Wilson, whereas Vermont (currently a socialist hippie paradise) being Calvin Coolidge country at the time).
Gun to the head, if I HAD to vote for a political party in Northern Ireland, I'd vote DUP, but I sure as heck wouldn't be happy about it. I don't want a government that loves the monarchy, loves being England's b----, and loves that Henry VII started a phony church so he could chop off his wife's head, and the aforementioned phony church now gets special rights and privileges while Catholics are treated as second class citizens.
Disclaimer: I do have Irish ancestry on both sides of my family. But my dad's side is culturally much more German and my mom's side is much more Italian.
I think it’s silly the Church of England is still established, and surprising given how secular the UK is now.
But I don’t really see how it would impact your life if you lived there, it’s of symbolic importance only.
And I don’t see how or in what way Catholics are/could be “oppressed” in today’s NI.