This story looks legit to me based on the individuals named in the article. Here is the info I found on them, in various issues of the Williams News and in James H. McClintock's history of Arizona:
Benjamin F. Sweetwood was one of the pioneers of Williams, Arizona , having located there in 1893 and attained prosperity as the area grew. He was extensively engaged in cattle ranching and was also the proprietor of the Sultana Theatre. He was born in Michigan in 1872 and arrived in Williams in 1893. He worked there for three years as bookkeeper for the Saginaw Lumber Company and then turned his attention to railroading for one year. He was connected for a number of years with the liquor business but then shifted successfully to cattle raising. Sweetwood's large cattle ranch was northwest of Williams. In 1912, he built the popular Sultana Theatre, a modern building for the time that could seat 664 people and had a roomy stage adequate for producing most any kind of play. Sweetwood married Nellie Seibers of Los Angeles in July 1902. Sweetwood was a member of the Williams city council an the Fraternal Order of Eagles (apropos).
George B. "G.B." Kirkpatrick was manager of the Postal Telegraph relay station in Williams, a prestigious role at what was considered an important hub of the telegraph system. Kirkpatrick arrived in Williams from Los Angeles in 1911 and made many friends due to his "gentlemanly and courteous treatment" of those who did business with the company. His family remained in Los Angeles although his wife eventually joined him in Williams. Kirkpatrick engaged in competitive shooting as a member of the Williams Shot Gun Club. He became involved in civic affairs and served as Williams' mayor before leaving for Santa Barbara in 1919 to establish an electrical and auto supply business. Kirkpatrick died there on March 25, 1949, age 81.
Bert Brown relocated to Williams as a child in 1906 (being in the 3rd or 4th grade). He married Mae Owen in 1916. Bert was working at that time for Williams Transfer and the couple settled in the city. In May 1918, he attempted to enlist in the U.S. Army (or was drafted, it's unclear) to fight in World War I but was ultimately rejected for service. The Browns stayed in Williams and raised a family.
Due to the cited wingspan of the eagle (over 8 feet), it sounds like a normal golden or bald eagle on the higher end of expected size. While this could just be one of many eagle trophy photos from the era, the fact that it took place in Arizona (Tombstone being center to the Thunderbird Photo legend) and could have been misremembered over the years as showing a larger bird (especially since the article said that boys posed with the bird) makes it worthy of investigation. The weekly Williams News had the ability to print photos in 1913, but frustratingly the issue immediately following the report on the giant eagle is missing from the archives. I'm putting out some feelers to institutions in that area that have historic photo collections. It would be very cool if this photo still exists! This is, I think, one of the only articles of this nature that specifically mentions that photographs were taken of the large bird that was killed. Not even the original Epitaph article (which is about a dragon-like monster, not a bird) does that, in spite of what people remembered. I will follow up if I find anything more.