First, your nice summary states, "if the function of marriage is the legal recognition of loving, or “voluntary intimate,” relationships ..." while the rest of the statement must logically follow, it does so vacuously ( p => q is equivalent to ~p or q ). It is not at all clear (and no evidence if given to suggest) that "the function of marriage" is as asserted. On the contrary, there is significant evidence that the function of marriage is the State's interest in regulating procreation.

Second, and I find this truly remarkable, the author(s) state, "The status of marriage itself confers legitimacy and invokes social support. ... Indeed, as the existence of intentionally childless married couples suggests, marriage has purposes other than child-rearing—notably, fostering a committed relationship." If the strength of any couple's relationship (regardless of sex) is in any way determined by the issuance of a State license with a specific word printed on it, it is a remarkably weak relationship. But this point must be made by same-sex marriage proponents because in its absence, assuming all legal rights, responsibilities and privileges are also granted civil unions other than marriage, there is no basis for the false claim of a "two-tier system."

Third, the author(s) write, "On his view, parenting is a basic civil right, the restriction of which requires the threat of a certain amount of harm." I feel like I'm in the film, "Idiocracy." Is anything, anything at all, not a "civil right" in this country anymore? Oh yeah, access to health care without funding Wall Street isn't a civil right. I almost forgot where I was living again.

Fourth, the author(s) write, "But social science literature shows that children are neither typically nor catastrophically harmed by same-sex parenting." The amount of study on this is far too limited to make the affirmative claim that "no harm" comes to children raised by "two mommies" or "two daddies" instead of the usual and customary "mommy and daddy." I make no claim that harm does come, but claiming the existence of sufficient evidence in its absence is textbook assuming facts not in evidence.

Finally, the author(s) write, "Nor do proponents of this objection to same-sex marriage generally suggest that entry to marriage should be restricted by excluding those unable to procreate without third-party assistance, or not intending to do so." Well, it is true that I've never advocated barring couples from marrying who are either incapable of natural reproduction or do not wish it, I still don't understand why anyone would want to marry in such circumstance. The strength of my relationship with my wife (31 years yesterday) has abso-fricking-lutely nothing to do with my marriage license. I don't give a tinker's damn what "society at large" thinks of our relationship and I do not depend upon third parties to define for me what that relationship is and what it means to the two of us. The two of us whom, after all, are the only two whose opinion on the matter is of any consequence.

The worst part of this argument is that there is some intangible benefit to having the State recognize my marriage as "a marriage." If Indiana were to draft a law that invalidated my marriage because the ceremony was not performed in a church (something that would not surprise me, btw), but allowed us to maintain all the legal rights, duties and privileges of marriage, would that somehow diminish my relationship with my wife? The idea is ludicrous. I wouldn't care and I doubt very seriously my wife would care either.

Edit:
In conclusion, let me hold up a mirror. You write that I can't prove I'm right or that my position is logical, well, neither can you. And I think you have the more difficult task. Children are a benefit to society, yes? Children are typically the result of heterosexual marriage, yes? So, there's the benefit to society of heterosexual marriage. Where and what, exactly, is the benefit to society of same-sex marriage?