“Polygamy works fine for producing children, so I don’t see that Natural Law prevents it.”
In fact, Natural Law probably encourages it. A woman is out of action while she is pregnant while the man who impregnated her can go on to impregnate other women. Our closest wild relatives, apes, are polygamous. It makes sense from a survival standpoint when infant mortality rates are very high as they are in the wild. I think maybe the traditional nuclear family makes sense when you see civilization begin. As man began to build permanent settlements and start engaging in trade and specialized labor a nuclear family is probably easier to survive in than a pack.
Men feel closer too and will be more motivated to work for and protect children that they know are theirs. In polyandry no man knows for sure which children are his and in polygamy men will take the wives of other men, this is not good for group cohesion.