Part of the reason gay marriage became legal in the US is because its opponents were unsuccessful demonstrating in court how such marriages pose any threat or harm to the community. What is obvious to religious believers (e.g. that a child raised by such a marriage is harmed in the process) is not obvious to secular society, and the religious failed to make a compelling case for their beliefs. So it stands to reason that pro-LGBT Mormons would employ similar reasoning as they lobby for change in the church.
Although the church is certainly not compelled to go along with any of this—the legality of tobacco, alcohol, and coffee have had no impact on the Word of Wisdom, for example—it does make sense to be able to articulate the moral philosophy by which homosexuality is deemed to be sinful. If it's just something we all know innately and never need question, then it can change as easily and as quickly as a new generation comes along with different ideas. For instance, if we know it's bad simply because it's disgusting, then what happens when we're replaced by children and grandchildren that aren't disgusted by it? Unless there is some solid moral reasoning for why it is a sin, they will be able to assume that we were simply misguided by our prejudices and they are more enlightened.
Moral foundation theory posits five axes by which we measure morality: care/harm, fairness/cheating, loyalty/betrayal, authority/subversion, and purity/degradation. While individuals judge actions according to all of these foundations, they are not a universal basis for public policy. We mostly agree on care/harm and fairness/cheating, and even loyalty/betrayal for the most part. There is less agreement about authority and way less when it comes to purity.
Notice that to a large extent liberals, moderates, and libertarians don't care that much about purity.
So if homosexuality is considered immoral solely because it is disgusting and degrading, that works so long as we all share that perception. But perceptions change. That's why the analogy to left-handedness is so apt. Long ago, it wasn't just considered a perversion, it was considered unclean. The right hand was used for eating and greeting and everything else, and the left hand was reserved for one's bathroom business. Like homosexuality, it made everyone think about butts. But that eventually changed.
So it makes sense to establish that gay marriage is intrinsically harmful. Or somehow unfair or a betrayal. Disgust is all we have needed so far, and that works as long as most people in society are disgusted by the same thing, but that can be fleeting.