The whole question of succession after Jospeh Smith’s death was definitely something of a mess. There were around a dozen men who claimed the right to succeed and Brigham Young was not initially one of the leading candidates.

To start off there’s no question who Smith intended to be his successor, that was his brother Hyrum. He had already laid all the groundwork and had Hyrum accepted and acclaimed (via a sustaining vote) as sharing most of his titles and authority. Unfortunately Hyrum of course died in the same shootout that Joseph died in. Likewise it was fairly clear that Jospeh Smith intended that his son Joseph Smith III should some day take over. But Joseph III was only eleven years old when his dad died. He certainly wasn’t in any position to lead the church. And he definitely wasn’t in a position to lead a church in crisis in a town under martial law and who feared they’d soon be in a shooting war with the neighboring towns and counties.

So it immediately became clear that one of the existing men in leadership should at least take on a caretaker role. The immediate obvious candidates were:

Sydney Rigdon. He was the sole surviving member of the First Presidency, and had been with the church since the Kirtland Days. Unfortunately for him he wasn’t particularly popular or well liked. He had been involved in the Safety Society scandal. Had not done anything well regarded in the Mormon-Missouri war, and had been increasingly sidelined by ill health. He also had fallen out with Jospeh Smith.

William Marks. He was the president of the High Council and was in charge of the church in Nauvoo. Which meant he was in charge of the vast majority (probably 80%) of member’s world wide. The High Council had nominally equal standing with the First Presidency and thus with the loss of the President the Council president should have equal standing. He almost immediately withdrew any claim he might have to support Rigdon instead.

Brigham Young. He was the president of the Quorum of Twelve Apostles. The Quorum also had equal standing with the First Presidency and the High Council. Plus Young was personally very popular. In the wake of the Missouri-Mormon War, Young had been one of the highest ranking leaders to escape imprisonment. He was the one who led the exodus into Illinois and directly helped thousands survive the winter securing food and shelter while Smith was in Liberty Jail. Plus as leader of the Apostles he had led the mission that converted thousands in the British Isles. In the years afterwards. So large swaths of American Mormons saw him as the man who saved their lives, while the 4,000 or so British immigrant Mormons saw him as the highest placed leader they personally knew and had met.

James Strang. Strang was only an elder, and a recent convert of a few months. But he had a claim because he had been set as the nominal president of an as of yet formed High Council in Wisconsin. To bolster his claim he produced a (likely fake) letter from Smith naming him as successor, and immediately began to claim to be receiving visitations from angels.

The immediate result was a plurality of members chose to follow Young and the bulk of the apostles. He was a known quality. He had shown an ability to lead in a crisis. And he was personally loved by most of the members.

A sizeable minority followed Strang, including most of the Smith family, a few apostles, and several other leaders including William Marks who had given up on Rigdon. These mostly consisted of the members who rejected polygamy, and thus Young’s leadership, as it was now an open secret that the apostles were practicing polygamy. Ironically Strang would eventually get his own polygamy revelation and marry several wives himself.

Rigdon gathered the rest, mostly the members outside Illinois include most of those that remained in Kirtland. A lot of these were remnants of the groups that rejected Joseph Smith himself at some point.

Young continued to do a heavy press on anyone who had been in leadership and had not followed him, offering offices and positions to any that would rejoin. The Strangite church collapsed when Strang was killed in 1856, and the Rigdonite church slowly withered. Each time something happened to one of the other Mormon branches, Young’s missionaries were there ready to point out how God was blessing the Utah church, while he had so apparently abandoned Strang etc. So by the late 1850s Young had captured the hearts and minds of most Mormons, and the mass migration to Utah was well underway.

And that’s when Joseph Smith III re-enters the picture. It was well known that Jospeh had intended for his son to eventually inherit the mantle of prophet. He had resisted calls for him to declare himself prophet for years. But now in his late 20s he finally felt himself ready, and with the Strangite and Rigdonite churches having collapsed, he saw a need to consolidate all the midwestern branches of Mormons who had resisted Young’s calls. So in 1860 Joseph III formed the Reorganized Church of Jesus Christ of Latter-day Saints. However Young had by that point a 16 year head start on the new Smith. Tens of thousands of Mormons had migrated to Utah, and Young’s aggressive missionary program had converted thousands of new members as well as continuously proselytized former Mormons.

Smith III gamely sent missionaries to Utah to proclaim the re-restored gospel and to denounce Young and polygamy. But ultimately he wasn’t able to gain too many followers. Young had already won the contest for the mantle of leadership, before Smith III even entered the match.