Without making a value-laden argument, how can I discuss how religions, as a key social institution, could take a different approach in order to enact social change concerning same sex marriage?
In answering this question, I assume that you mean to ask how religions could cause people to stop approving of same sex marriage or how they could make it become illegal again.
At this point, it is not clear that religions as a group could actually bring about social change on this issue. Religions have been an important social institution and have been (mostly) opposed to same sex marriage and yet have been unable to persuade enough people to oppose gay marriage. It may be that churches are simply not important enough of a social institution anymore to be able to move public opinion on this topic.
However, let us assume that it is possible that churches could bring about social change on this issue. One possible way to do this would be to decouple the religious aspect of marriages from the secular. Churches could simply stop having their wedding ceremonies be legally valid. People could get married in a church if they wished, but that marriage would not be legally binding. If they did this, they would also have to fill out some sort of a government form in order to be legally married. In this way, churches might be able to impart the idea that same sex marriages are something that the state recognizes but that the church does not. This might possibly get some people to think differently about the issue because it would separate the religious aspect of marriage from the secular, legal arrangements. People might then stop feeling like it is unfair to deny same sex couples the right to marry (in this case, to marry in a religious ceremony).
Another way that churches might possibly bring about social change would be to push for truly traditional definitions of marriage. Some people today may feel that the churches are being hypocritical on same sex marriage because they say that same sex marriage is abhorrent while accepting opposite couples who marry multiple times and who do not seem particularly dedicated to the idea of marrying for life. Churches could conceivably make a more principled stand in favor of tradition marriage and against both same sex marriage and opposite sex divorce. This might convince more people that the churches really do value traditional marriage and that they do not simply dislike gay people.
Finally, it may be possible for churches to change people’s minds if they become more gay-friendly on other issues. As it is, churches often tend to look as if they are anti-gay. This may be causing a backlash in which people support same sex marriage because it seems as though the churches are opposing it out of bigotry and hatred. If churches could more convincingly show people that they value gay people in general but do not approve of them getting married, people might think better of the churches and might be more willing to accept their arguments against same sex marriage.
I am not at all sure that any of these things would actually work to get people to oppose same sex marriage. People seem to be moving away from respecting the churches’ vision on social issues and it may not be possible for churches to win them back.