It may be that not all mid-19th century American amateur backwoods "flower namers" were as meticulous with Latin as you are, Catamariner. Someone may have taken "aro" (basic first-person singular present form of "to plow") and "natus" (basic nominative singular perfect participle of "born/spontaneously produced") and just stuck them together. As far as we know, the word never made it into any published botanical work.

— the word in the sheet music —