Actually, there's no particular reason that a Church should not try to influence politics. The whole "Seperation of Church and State" thing is only something the government has to avoid. For example, suppose a proposal on legalised gambling is put up for vote. When a particular person votes they can do so because they simply dislike gambling, or because they think it's bad for society or they think a person should have the right to do whatever crazy thing they want as long as it doesn't hurt anybody else. The same thing goes for an organization. They can bring the issue up to it's members and request them to vote a certain way. This doesn't destroy our system of government. If a particular vote approves something that infringes on someone's rights or civil liberties then other branches of the government can overturn it. The whole point of the US Government is to balance the will of the people against mob mentality. The idea that as a non-profit group you're not allowed to try an influence politics is silly.

You're basically advocating a kind of "Your religion is fine, just don't let it get in the way of real life" view. To Mormons, the Church is their life, and they'll do what they can to keep the world around them from becoming unbearable. The LDS Church isn't trying to persecute gays, it's just supporting its (and most peoples) view of marriage. Supporting the view that homosexual sexual activities are sinful doesn't make you anti-gay. In that case, the LDS Church is also anti-coffee-drinkers and anti-excessive-meat-eaters. Uh-oh, I fit in that last category. That doesn't stop members of the church from running around doing all kinds of stupid things though.