The Church’s official explanation for why they would do something like this is that they are protecting children from the uncomfortable situation that might arise when the child is on the records of the Church, but is not supported at home. See the official response here (by the way, notice how the interviewer hand-feeds these gentle and obviously scripted questions to Elder Christofferson).
I find at least three concerning issues here. The first is that it appears that the child is being punished for having same-sex parents, not protected from discomfort. Second, the Church seems inconsistent with this bizarre practice of barring a child’s alleged spiritual progress based on the parents’ beliefs. Lastly, why is gender more important than love?
First, the claim is that this policy protects the child from conflict. I cannot understand how the policy would accomplish that. If a child wants to be a member of an organization that believes his same-sex parents are committing a very serious sin, the conflict is inescapable – it will not go away at age 18 or ever. At best, this Church policy only makes the conflict worse by forcing an immediate choice – “Do I want the blessings of my chosen faith, or do I want to live with my same-sex parents?” Had the policy not been in place, a child whose same-sex parents did not oppose his decision could be baptized, go on temple trips, and so on. Now that it is in place, the child is forced to choose between his same-sex parents and his faith, even if the parents have no objections to his membership. Rather than reaching out to the same-sex couple by showing acceptance to their child, the Church has chosen to cut off the family for as long as the same-sex couple is an issue.
Second, I find this policy wildly inconsistent. Why implement this stance in families with same-sex marriages, but not in other families that do not fit the LDS ideal? For example, suppose a liberal LDS couple (yes, they exist) teaches its children that same-sex marriage is okay, even though the Church does not condone it. Why would their children be allowed baptism, if we are using the reasoning provided by Elder Christofferson? If the aim is to avoid children being taught things in the home that are contradictory to doctrines, shouldn’t thousands of children be denied baptism each year? Indeed, it sounds like the next question for the temple recommend interview should be, "Do you here and now condemn same-sex marriage as a sin comparable to murder?" I'd wager that a good proportion of members, particularly of the younger generation, would not pass the interview.
Lastly, I am an enormous supporter of the family. The family has become a hot topic in recent years as opposing views wrestle over what the family is and isn’t. On the one hand, some want to portray it as perfectly normal for children to be from broken homes, raised by nannies, by helicopter parents, and so on. On the other hand, others, like the LDS church, seem to suggest through policies such as this one that the family is primarily about genitalia, and the rest is secondary. The Church wants the world to think that children raised in same-sex households are in grave danger, simply due to the fact of the same-sex union. However, I believe that what is more important in a family is that the parents are unified in their love and support of the children: nurturing their interests and talents, and guiding them through life’s tough times. I argue that such characteristics are far more important than the sex of the parents. Perhaps same-sex parents are not the ideal situation, but if the child is loved and nurtured, then that's a far better situation than many children have whose parents are straight.
Naturally, as society becomes even more accepting of same-sex unions, and science continues to demonstrate that it is not an issue of choice, morality, or sin, but a mystery of nature, I am certain that the LDS church will be forced to follow the path it took regarding race and worthiness – either face mutiny or change doctrines. If history is any indicator, the Church will again choose the latter.