Natural Law is used by Catholic Theologians all the time—and in profound ways.
The reason Natural Law prevents polygamy is that it reduces a woman to cattle and unequal to men. Christian Ethics were embedded into Natural Law by St. Thomas Aquinas in 1245 and the Natural Rights of children to be raised by their own biological parents (only) is actually a basic component of the Laws of Nature.
All children have the Natural Right to be raised by their biological parents. Any system that denies this “right” reduces children to a commodity to be bought or sold. It is dehumanizing and should be unconstitutional. It uses human beings as a “means to an end”—like slavery. It is immoral. Laws have to be moral to be “Just”, since Justice is a Virtue. There is no such thing as an immoral “Just” law. Promotion of Vice is unconstitutional unlike what Justice Holmes stated, whose socialism/progressivism destroyed our “Justice” system.
Polygamy puts in a system where women are not equal to men. The only system which guarantees “equal rights” to all involved—even children is the one-man, one-woman system which created the most free successful country in the world.
"Natural Laws are basically Common Sense. Cause and Effect/Laws of physics, biology, human design and needs"
Human biology, as another poster pointed out, makes Polygamy viable, and perhaps even optimal for creating the most offspring. You say natural law is opposed to polygamy, though:
The reason Natural Law prevents polygamy is that it reduces a woman to cattle and unequal to men
So "Natural Law" - based on cause and effect and common sense - insists that Men and Woman are equal? That seems strange and wrong. Men and woman are *not* equal in almost anything. Men are stronger, faster, more aggressive, etc. Woman have their strengths, too, obviously, and carry babies for 9 months when they are pregnant.
Wikipedia's definition of Natural Law is much like yours: "certain rights or values are inherent in or universally cognizable by virtue of human reason or human nature"
I'm not sure how Natural Law guarantees "equal rights to all"... That sounds pretty modern.