Many of the people appointed seem to only have lasted less than a year or two, especially during and just after the Civil War. Why is that?
The primary reason for short terms was that most governors couldn't thread the needle of keeping the Feds, the Mormon population, and non-Mormon population happy. Cumming was escorted by an army and relied on it to keep him in power for four years. He was replaced by Dawson who lasted a month before he was savagely beat by three Mormon men, who were later killed during their arrest. Harding then replaced Dawson, but only served for a year before Lincoln had to replace him due to constant petitions against him from the Mormons.
He was replaced by Doty, who set a new precedent by dying in office. This new pattern was followed by Durkee who replaced Doty. And while he didn't die, he did get sick and Mann had to serve out the rest of the term. He was replaced by Shaffer who made it a year before getting sick and dying. Vaughn was then appointed to finish the term.
Vaughn put the men of a Mormon militia, the Nauvoo Legion, on trial for rebellion for holding what he called illegal drills. That completely poisoned his relationship with the Mormons. So Grant didn't re-appoint him. But at least he survived his term. Woods became the new governor. He constantly fought with the Mormons, the Nauvoo Legion, and the probate courts. So he didn't get a second term. But at least he filled a full term. He was the first one to get a full term since Cumming. Axtell got the job next. He worked hard to keep the Mormons happy. And in so doing, pissed off the non-Mormons. So he didn't last a year before Grant had to transfer him to New Mexico.
Emery then got the job. And he was probably the only one who managed keep everyone happy. But he only served one term. The job by then had became more of a political reward. So each president started making appointments of their own men, regardless of how well they served. While the governors didn't get more than a term, except West who got 2 non-consecutive terms. They at least served full terms.
For more details you can check out the Utah History Encyclopedia. Territorial Governors, by Miriam Murphy.