In his 1916 book “Fear God and take your own part” Theodore Roosevelt wrote that if Arthur were to go to England, he could have been pressed into military service. Clearly Roosevelt knew Arthur had dual citizenship.
It appears Chester Arthur was eligible for dual citizenship, meaning that another sovereign could have claimed his allegiance. The United States pays no attention to such claims. It might affect an individual's travel plans, but not his presidential eligibility. Today, an individual may have to be careful about flying into the airspace of a nation that can claim him.
Roosevelt goes on to describe the case of a Mr. LeLong who was born in New Orleans to a U.S. citizen mother and an alien (French citizen) father. In describing Mr. LeLong, Roosevelt wrote “He is eligible to the Presidency of the United States.”
Roosevelt was right. Mr. LeLong was born in the United States, subject to its jurisdiction. He became a U.S. citizen at birth. The status of the parents is irrelevant. That France might claim him as a citizen of France makes no difference. If he actually held French citizenship and traveled to France, he could be pressed into the service of France. If he were President or Vice President, that would not apply. When not a dignitary, it might affect his travel plans, but not his presidential eligibility.
Congress did not pass a law requiring two citizen parents. Congress proposed an Amendment to the Constitution. The People ratified the words proposed for ratification. The words that were ratified did not include anything about parents.
Arthur born in 1829 to a father who was a British subject would have automatically been a British subject under British law.
I suspect that this would be the case even had his father naturalized as a US citizen before Arthur’s birth. England still recognized perpetual allegiance until 1870.