I don't think that's accurate at all. Government has been involved in marriage in this country since at least 1629, when the Massachusetts colonial legislature passed its first law governing divorce. Even earlier if you consider the legal treatment of marital property, or common-law dower and curtesy rights for surviving spouses.
And during and before that time there was an established Church, and so government could not "intrude" into the "sacred arena of religion," because there was no dichotomy between the two.
Just due to marital property rights alone, I don't think there is any realistic scenario where government does not have some involvement in marriage.
Yes, but the rest is after the fact, dealing mostly with property division. It was the Church, in Judeo-Christian tradition, that authorized and sanctified marriage. It probably varies from culture to culture. But in Christianity, it is a religious rite, not a legal right.
The Jim Crow licensing of marriage by the state was a critical turning point in US legislative history.