It could be Greenlee, AZ, then. AZ was after all not made a state until 1912 and 1912 was its first presidential race, so if Greenlee was made at the same time as AZ and wasn't made later on, this would track with the part about the GOP coming in 3rd in 1912. I also know Greenlee was an ancestrally blue county that voted even for McGovern in 1972 (though not entirely sure if it went for Stevenson in 1948 and 1952 and if it was blue prior to the New Deal), but which has not gone blue since 1996.
Greenlee County, Arizona, is the correct answer.
However, I made a mistake, which I've already corrected; Taft was apparently so unpopular that he even fell behind Socialist Debs. lol.