Fascinating! I'm sure there is much more to do to further understand how religion impacts suicidal ideation and such, but this certainly looks like one entry level piece on the subject. Almost certainly not the last word (I think the authors themselves say as much) on the subject. A few non-expert reactions to the opinion piece and the BYU paper:
1) The BYU authors emphasize early on that "significance" in the paper means that they were able to conclude "statistically significant" from their statistical tests. "Significant" does not necessarily say anything about the real world "size" of the effect. In other contexts, I have seen some criticize "high n studies" (where the conclusions are based on a large number of participants -- I don't know if n=86k is considered large n for this kind of work). Because of the large n, the criticism goes, the study has strong statistical power to "see" small differences between groups, but the perceived differences are still very small. At some level, even if the difference between LDS and other religion or no religion is statistically significant, is the difference large enough to have practical meaning?
2) As with anything like this, there is always the "correlation does not mean causation" thing going on. The authors find a statistically significant correlation between checking the LDS box on a form and less suicidal ideation/attempts, but that does not mean that being LDS prevents suicide. I expect that a large part of the future work that wants/needs to be done is trying to understand what factors drive the correlation.
3) I appreciated their attempts to address how disaffiliation might confound the conclusions. I can't say that I understood everything they did, but it does seem like an important thing to include in this analysis. It was somewhat gratifying that, even using their best guesses at disaffiliation numbers, the final conclusion did not change. However, in their discussion of disaffiliation, they also note that, assumptions along the extreme end of their alleged uncertainty limits, could change the conclusions, so their appears to be just enough overall uncertainty to claim, at the outside, that maybe some of the paper's conclusions are because the SHARP data do not include any indication on disaffiliation. On a personal note, perhaps just because of where some of my own thoughts are on the topic, if identifying as LDS is somehow correlated with lower suicide rates, is there some way we as a Church can do something more to discourage disaffiliation?
I do not have the expertise to provide any expert opinions, but it seems like a good entry into the discussion. I look forward to more data to help clarify the relationships between the Church and its LGBQ members.