I'm fascinated by the degree to which Postmillenialism dominates Christian thinking. I would presume Mohler is a Premillenialist of some sort (being Baptist). At most, amil.
We must understand that, fundamentally, the majority will always reject God ("the wide road"). The founders of America were not strong Christians, but they did operate under the assumption of a Christian worldview - and that the population would continue to hold that view.
We see now, the consequences of such a system; where the population ceases to hold that view.
Why does the State involve itself in marriage?
From a "Christian nation" point of view, it is to encourage the formation of families. Tax breaks (in the form of tax tables, and benefits like health care) allow for one parent (originally the man) to provide for a family. Single people complain of unfair treatment, and the "marriage penalty" is born. Other groups seek State sanctification of what they call marriage - in order to access the benefits.
"There is no major society that exists without marriage, and those rare movements in history that sought to eliminate marriage led to disaster."While true, we cannot operate from this motive (pragmatism, or even "the ends justify the means").
We are seeing the fallout of where a Christian derived system is reconciled with the majority rejecting Christ. The question is: can a stable (and God-honoring) system be derived where the majority reject Him?
I don't know, but I don't know if anyone else is even asking the question.
Some are calling for the State to withdraw from involvement in the regulation of marriage - and I must agree (and disagree with Mohler's claim this "would lead to legal, moral, and cultural chaos"). There are standard forms for wills, etc. which are not blessed by the State. Christian men will continue to provide for their families as best they are able. Non-Christians will continue as well as they are able.
But this is just removing a vestigial remnant of the Christian worldview. It does not address replacing and fortifying underlying structure.