It is Mormonism, Mohammedanism and heathenism and not Christianity which have proclaimed polygamy and debased woman from the sacred place of wife to the lower level of concubine. It is not Christianity which has sustained the social evil.
Society may no longer define marriage in the only way marriage has ever been defined in the annals of recorded history. Many societies allowed polygamy, many allowed child marriages, some allowed marriage within families; but none, in thousands of years, defined marriage as the union of people of the same sex.
The state's interest in marriage is stability. Generally speaking, polygamy does not work for stability. Inherent in the whole polygamous movement is a deep and abiding misogyny and denigration of women. So polygamy is objectionable on lots of grounds.
The government doesn't really prosecute for polygamy anymore, but a lot of the arrests are of groups supporting themselves through welfare scams or for child abuse. So that was all I'd really heard about polygamists.
No Congress ever has seen fit to amend the Constitution to address any issue related to marriage. No Constitutional Amendment was needed to ban polygamy or bigamy, nor was a Constitutional Amendment needed to set a uniform age of majority to ban child marriages.
Polygamy is needed to prevent a man from taking up a lover, which can cause health issues and rifts in the family and would lead him to live a lie, constantly deceiving his wife and children.
The principle of plural marriage was revealed to the Mormons amid much secrecy. Dark clouds hovered over the church in the early 1840s, after rumors spread that its founder, Joseph Smith, had taken up the practice of polygamy. While denying the charge in public, by 1843 Smith had shared a revelation with his closest disciples.