One could be born an Irishman, and still be born in the US, of parents who were US citizens. Since Arthur's mother was born in the US, and thus a citizen, and his father was a citizen by the time he became President, no one thought to ask *when* his father became a citizen. As it turned out it was 14 years or so after Chester's birth.
Being an Irishman can be just a state of mind. Lots of Irishmen fought for the Union, and for the Confederacy. Their grandchildren often still thought of themselves as Irish. (I had an Uncle, by marriage, like that He was James, his brother was Patrick, they went to St. Patrick's church) He would have been born around 1910 or so, but it wasn't in Ireland, but rather in Nebraska, and I don't think his father's birth was either. But he could turn on that brogue at will.
I once knew a guy, a least third generation US citizen, that spoke with a German accent. Dude looked like slightly taller and think Hitler. He was President of the campus Young Americans For Freedom. The father of my doctor, was a south Texas Rancher, who would not speak English in his home, although outside it, he could speak it just fine.)
Being Natural Born doesn't completely guarantee one would not have sympathies for a foreign nation or people, but making that a requirement for eligibility to the office of President was about the best they could do.
That's exactly correct. It's not a 100% guarantee, but compared with someone born with foreign citizenship, or influenced by a foreign national parent(s) it does increase the chances of not having divided loyalties, owed to a foreign state.