In 1870 Mormon leaders were concerned with three key things that they worried would lessen their influence and control over Utah politics. The first was the completion of the transcontinental railroad the previous year. Brigham Young spoke frequently of "outsiders" coming into Utah and having a corrupting influence. He worried that these would be single young men coming to the territory to mine or to find other work. These men would be able to vote and influence Utah elections.

Second, a small but influential faction of Mormon businessmen was opposing Young's cooperative economic plans. Young wanted Mormons to buy from other Mormons, so faithful members were given the seal of approval, as it were, to sell goods. They were expected to meet certain standards as part of this cooperative movement, but others objected to some of the rules, especially the proscription on importing goods from "outsiders" to sell. Young wanted Utah and the Mormons to be self-sufficient and wanted goods produced within the territory. Young during this era spoke almost nonstop about the evils of buying goods from "outsiders" or from unapproved sellers. Those Mormons opposing him became known as the Godbeites, after William S. Godbe. There's a lot to the group, including an embrace of spiritualism and seances, but they represented an organized movement against Young. Most of them were excommunicated by the LDS Church. These same people would found what became the anti-Mormon Salt Lake Tribune to promote their interests. They, with other non-Mormons, founded the Liberal Party, the first serious political party not dominated by Mormons or LDS leaders. It took years, but the Liberal Party made consistent inroads and would eventually win elections outright.

Finally, Mormon leaders worried about a growing movement to disenfranchise polygamists. Although the most prominent anti-polygamy legislation did not pass until the Edmunds Act of 1882, it was a constant threat within the US Congress. By 1870, Utah was also led by non-Mormon federal appointees from the East. Judges like James McKean led anti-polygamy crusades and territorial governors gave anti-polygamy speeches to the LDS-dominated legislature. The 1874 Poland Act would give teeth to the mostly benign Morrill Anti-Bigamy Act of 1862.

All of this added up to Mormon leaders needing as many pro-Mormon votes as possible in the Utah territory. LDS support would lessen the influence of men who traveled to Utah on the railroad, it would blunt the efforts of the Godbeites and the Liberal Party, and it would ensure Mormon control of elections even if polygamist men were disenfranchised. They accomplished this in two ways: First, they encouraged the thousands of immigrants arriving from Europe to become citizens so they could vote. They gave classes and provided information on naturalization and on owning property. Those who had claimed property now had to go through the process of proving it was theirs to the federal government. The second way they tried to ensure continued LDS control of elections and local offices was by giving women the vote. The Utah Territorial Legislature passed the law in February 1870. Ironically, the Godbeite movement was one of the primary pushers of women's suffrage in the state. The Godbeites hosted national suffragist leaders like Susan B. Anthony and sponsored lectures with them.

Women in Utah territory, contrary to the many dime novels and popular depictions of them as trapped and victimized, were supportive of their church. Their relationship to polygamy was complicated, to say the least, but it was not universally abhorred. For some women, polygamy offered an escape from abusive, alcoholic husbands into a large household of supportive sister wives and a respectful husband. Certainly for many other women, including many of Brigham Young's wives, polygamy was a lonely, often unpleasant experience. Some of Young's wives lived in poverty in other cities while his favorites lived in massive homes (Amelia Folsom was the hostess of the Gardo House). But regardless of their messy relationship to polygamy, most Mormon women were not supportive of federal efforts to tamp down on the practice or to imprison Mormon men.

Mormonism also offered women certain freedom within their own institutions. The Relief Society was largely independent, with its own magazines written and published by women. Women founded the Deseret Hospital and sat on its board of directors. Prominent LDS women like Emmeline B. Wells, Eliza R. Snow, Bathsheba W. Smith, and many others, were influential in Utah politics and its civic organizations. Mormon women, in other words, were not pawns of LDS leadership when it came to the vote. It was something they wanted and used in their own way and with their own influence.

Giving LDS women the vote would guarantee support for Mormon politicians and would help LDS leaders retain control of local offices that were not subject to federal appointments, such as mayors and city councils. Suffrage ultimately helped, especially as men were imprisoned for polygamy following the passage of the Poland Act and the Edmunds Act. The Edmunds-Tucker Act of 1887 rescinded women's right to vote in Utah territory by requiring a test oath of Utah citizens. The right to vote was restored in the state constitution, enacted upon statehood in 1896.

On Brigham Young and cooperation, see Devery S. Anderson, ed., Salt Lake School of the Prophets (Salt Lake City: Signature Books, 2018). On the Godbeites, see Ronald W. Walker, Wayward Saints: The Godbeites and Brigham Young (Urbana: University of Illinois Press, 1997). On anti-polygamy legislation, see Sarah Barringer Gordon, The Mormon Question: Polygamy and Constitutional Conflict in Nineteenth-Century America (Chapel Hill: University of North Carolina Press, 2002), and Edward Leo Lyman, Finally Statehood! Utah's Struggles, 1849–1896 (Salt Lake City: Signature Books, 2019). For more on women and polygamy, see Laurel Thatcher Ulrich, A House Full of Females (New York: Knopf, 2017), and for Brigham Young's family, see John G. Turner, Brigham Young: Pioneer Prophet (Cambridge: Belknap/Harvard University Press, 2014).

That's just a fraction of the literature on this. While studies of Mormon polygamy has historically been very male-centric, especially around church leaders, there has been a lot more focus on women and how they negotiated the practice. There have been some wonderful dissertations and journal articles on this topic.