The only way I could see this working is if Mormons never mostly become a single, very organized group but instead metastasize with radical protestants so that what we would call evangelicals or pentecostals are in this timeline pejoratively called Mormons, and there's a huge number of groups with a diverse set of beliefs that are called by that name.
Radical protestantism identifies itself with the principle that the bible (usually King James version to the exclusion of others) is the first and last word in the faith. Statues, paintings and rosaries sported by Catholics are seen as unnecessary, if not offensive to their fundamentalism. So, the notion of another testament (Book of Mormon) would classify the Mormons as non-Christian, a notion today shared by many Baptists and evangelicals.
So, to pull converts from other churches, they would have to do the work individually, as in OTL.
Frankly, all you need is massive proselytizing. That's how religions grow. In the scenario, Mormon missionaries simply preached to so many people across America throughout the second half of the 19th century, a very large number converted, had many children, and by the early 21st century, a third of the American people were members of the LDS Church. It's as simple as that.
That is how the Mormons so thoroughly settled so many Rocky Mountain regions. In fact, their OTL success is quite impressive, though coming nowhere near 1/3 of the population.
To gain a massive wave of conversion, the Mormons would need evidence: as I said in post #11, they might produce artifacts that can be linked or associated with the plates/foils Joseph Smith described as he founded the faith. Forgeries might work in the 1800's, but to be successful in the twentieth century, the artifacts would have to be authentic. With such artifacts, they might make inroads with Latin-American Catholics and Native Americans, on the notion that the faith says Christ came to America. Converting Catholic bishops might be far-fetched, but not impossible.
The temple getting burned down was an effect, not a cause, of the Saints leaving.
Yes, the migration started while the temple was being finished. But the Mormons would like you to believe it was the cause of their flight. [They do have a knack for telling history their own way.] In any case, I would think some would have stayed if the temple remained intact.
The main reason they were driven out of so many places was that they didn't police their followers. Some of their members did not respect other people's rights; and it was their actions, not beliefs, that caused their neighbors in Ohio, Missouri and Illinois to drive them out.
Once they left Nauvoo, their reputation for responsibility changed. Suppose they became more respectful from the start, eventually leaving settlements and temples from community to community.