The actual motive for having governments recognize the union of a man and a woman (and only such a union) as a marriage is to encourage, in a gentle and non-coercive way, the formation and maintenance of a stable environment in which children can naturally come to be. If heterosexual coupling did not regularly produce children there would be no reason for the institution of marriage to exist, let alone for governments to recognize it.
What a depressingly -- implausibly -- narrow view of marriage.
No doubt, children are a common byproduct of heterosexual marriage. That's certainly been the case in my marriage, and I'm glad of it. But the pairing instinct -- one that predated any government recognition of the "institution" of marriage -- far exceeds simple propigation of the species. People, as a general rule, want company. They want sex, they want economic partnerships, they want somebody to hang out with.
To reduce marriage to merely a mechanism of natural child-creation -- as National Review and other conservatives regularly do, because it's pretty much the one thing that heterosexual marriage offers that same-sex partnerships can't -- is, when you think about it, a surprisingly Darwinian argument coming from a movement that is largely theology-minded. It aggressively ignores that humans are social, spiritual creatures and that they express those characteristics, often but not exclusively, through marriage. The conservative case against same-sex marriage reduces the "institution" to simple biology. It's a point of view that reduces humanity to the level of beasts, with a bureaucracy.