If one wished to come up with a taxonomy for different ways of looking at marriage, one might call the view expressed in the Cotton excerpt the “social solidarity” view of marriage. That is, it does not approach marriage primarily from the perspective of my fulfillment, my happiness, my sanctification, my wishes, etc.–what one might call the “self-cultivation” view of marriage. Nor does it approach marriage in abstraction from society as a whole–what one might call the “isolated family” view of marriage (work with me, I’m making this up as I go along).
Rather, Cotton approaches marriage in the first instance from the perspective of one’s objective duty to one’s fellow men, that is, as an outworking of the law of nature and of God’s intention for the human race, and this is why he begins with the propagation of the human race, followed by, more specifically, the propagation of a holy seed. This is not to say that he denies some of the other (subjective) things mentioned above; but he does not begin there. He begins with children (i.e. the furtherance of humanity), something much more concrete, I think all would agree, than “fulfillment,” “desire,” and so on.
What is the cost of beginning the other way around, proceeding from the subjective to the objective instead of making one’s start with nature and creation? A brief look around at the confused question of marriage in the West will give you the answer forthwith.