Since it was being publicly alleged that Chester Arthur himself was born in Ireland, it must have crossed more than a few minds that his father was still an Irish citizen.
I said, "known", not "speculated". I'm sure it did cross a few minds, but they were so intent on finding the "foreign birth certificate", or at least proof thereof, they didn't look for naturalization papers for his father. Of course since Arthur actively concealed the fact, missing only the copy in the National Archives, even if they had looked, they might not have found it.
There were really no "birth certificates" as we know them today in those days. Births, and marriages for that matter, were recorded in the family Bible, or equivalent, and in church records. Churches without infant baptism might not have a record of place of birth either.