The revelation that the Mormons were practicing polygamy more or less ruined their early missionary efforts. Remove that aspect and they're still going to head West, due to the fact that they pissed everyone off by being a really disruptive group every time they settled down. But they'd be able to make it to Utah and still be boosting their numbers drastically, though I'd question how much further they could really go with that generation's missionary work.
Doesn't solve all the problems with the AHC, but it does help with the primary issue of population.