My $0.02.
You ask for proof that allowing gay marriage in some way confers some benefit to society as a whole. The current argument for why we even bother to recognize and reward heterosexual marriage seems to be based around the inherent stability that a two parent household creates for children. Furthermore, the argument appears to also be that this stability promotes procreation in a controlled setting. I would argue that allowing for gay marriage fulfill both these conditions for a significant subset of the population.
First, I would argue that the pursuit of equality among citizens is a worthy end in itself. A major measure of the virtue of a civilization is the equal freedoms (and thus rights/rewards) enjoyed by citizens. You argue that a gay man has the right to marry or not marry a woman like any straight man. If, as I believe and science increasingly supports, homosexuality is not a choice and is a largely genetically/otherwise physiologically determined state, this is not equal. Genetically, they are predisposed to find the prospect of a close romantic relationship with a woman unsavory and unfulfilling. Imagine a tax break or subset of rights that existed for only people who ate gluten. Somebody with celiac’s disease (a genetically/physiologically coded state) COULD choose to suck it up and eat gluten, but they would suffer immeasurably for this choice.
Furthermore, as medical/reproductive science progresses, the ability for gays to “procreate” increases. I would not doubt that within our lifetimes it will be come increasingly common and possible for genetically male/male or female/female children to be birthed. Beyond that however, is a lesbian couple who uses donor sperm and IVF any less worthy of a child and having the support than a heterosexual couple who goes through the same process? And before anybody notes that this isn’t technically 100% their biological child, should children born of affair have their parents’ marriage benefits revoked? How about a child via adoption? Giving the same benefits to a homosexual couple that is interested in raising a child confers the same benefits as giving those benefits to a child rearing heterosexual couple.
In regards to how this benefits society, I would argue that supporting the union of a homosexual couple confers many of the same important benefits supporting the union of a heterosexual couple does for the society. I just went over why it confers the same family stability benefits as a heterosexual union. Beyond that however, promoting a union between two individuals promotes a more stable society beyond the family setting. Somebody with a significant other through their life is going to be inherently more stable than somebody who does not. They have something to lose. It is easy to argue that you can do this without a title or benefits, but it is worth noting that there are many marriages that would not exist for the length they do if not for some of the psychological (and legal) hurdles required to dissolve the marriage. Signing a marriage contract suddenly seems to make the relationship worthy of a lot more work before dissolution. This in a way ensures that more individuals will keep at least that one other person around who can make them feel more connected on a personal level to society and the world as a whole. Somebody who is utterly alone is far more dangerous to a society than somebody who is married to someone, regardless if they are still madly in love or simply get along (non-romantically) but are not in such a way they wish to go through the new hurdles required to dissolve their relationship.
In regards to why gay marriage should be recognized and not polyamorous relationships, this is truly comparing apples and oranges. The dynamics in a two person relationship (stability, etc) vary drastically from those in any 3+ person relationship. In a two person relationship, there is equality in the commitment given-those two individuals are (in theory at least) completely committed romantically to the other individual in the relationship. It is a binary relation-1 to their partner, 0 to all else. With two, three, etc partners, suddenly this becomes incredibly more complex. No longer is it in shades of black and white. By splitting the commitment, you are opening an entire world of shades of grey. Therefore, the gap between the relationship between these partners and those they share with others outside the relationship is incredibly thinner than in a two person relationship. Thus, the inherent stability of the binary system is not present.
Also, the problems with governing using the lens of religion is that there is indeed so much disagreement. In regards to Sharia law, 'Murica style, goes-there are plenty of aspects of Biblical law that would be abhorrent beyond the establishment of Christianity as the official religion. I have no problem with people using the Bible as a personal moral compass, if they need that to define their morals so be it. However, saying something should be a law because “the Bible says so in Name:number:number” is a slippery slope to the Old Testament.
tl:dr:
-Through adoption or other means, homosexuals can have children. Supporting monogamous relationships in these cases creates stability.
-Even in the absence of children, society benefits by the coupling of its members.
-Polyamorous relationships are non-binary and thus far different than the binary romantically committed/not romantically committed relationships that lend stability.
-The Bible, while great as a moral compass, should not be used as a reason unto itself for laws.
This was long, I did not proofread it thoroughly, I am headed out-I tend to ramble so hopefully this was coherent enough. If not I will edit later.