I don't know when government got into the business of licensing marriages, but I'd wager it was in relatively recent times perhaps the mid-to-late part of the 19th century. Before then, most marriages in most societies were based in religious and cultural traditions, i.e., the private sector.
It is also the method of preserving the values of the society and passing those values on to the children. It is the only reasonable way short of the "Brave New World." Without traditional marriage our society will collapse. Another society will take its place. I doubt it will be good.
Shalom.