(1)Sexual relations between a man and and a woman are the ONLY kind of sexual relations which can spontaneously give rise to the existence of another human being. Securing that child's rights to the support and nurture of both his natural parents --- father and mother --- which is the essential public purpose of legal marriage.
(2) Although there are man/woman couples who are either unable or unwilling to have children, it goes beyond the scope of government of surveill their marital relations and/or their medical records to determine whether they do or don't intend to remain childless.
(3) Therefore the state, to secure a child's natural rights without invading the marital privacy of the couple, simply assumes that every man/woman sexual alliance is potentially procreative, and thus is eligible for such measures as the state may take to augment its stability.
(4) Why wouldn't this be convincing? It's been convincing for 3-4,000 years. It's not perfect--- not by a long shot --- but man/woman marriage does involve a public interest that flows from its one unique characteristic: fertility.
Why would the state want to license love and affection? ~We're~ interested in love and affection, but the state's interest isn't romance: it's in the protection of the procreative unit, and securing the rights --- very much including the property/economic rights ---of the progeny.
I’ve read that romantic love is a recent focus (but see the Song of Solomon). That prior to the romantic era marriage was much more of a familial, social, and financial/material bond. So we’ve sort of gone from a poetic overreach regarding male/female relationships to using that overreach to justify homosexual marriage.