Part of me admits a superficial attraction to this idea, though I don't think it's workable in the long run. After all, as a reader of history I know perfectly well that the fusion of secular and religious (Christian) marriage is only about a thousand years old. In the early Middle Ages, families concluded marriage contracts (to which the individuals consented). The Church got involved by blessing marriages of Christians at the church door after the contract was signed. Eventually, a more or less standard wedding liturgy was developed, but it was a much later arrival. So, sure, there is no essential union between the contract and the covenant that both go by the name "marriage." On the other hand, you have only to look at the attempt by the Roman Catholic Church to maintain a separate canon law definition of marriage -- one that does not officially allow divorce, and which says that some marriages are canonical and some are not -- to realize the difficulty of even so strong an institution as the RCC maintaining its definition of marriage against the pull of the larger society's less rigorous definition.
Still, it's an idea that bears examining. What is the interest of the State in defining marriage? And what is the interest of the Church?
Well, the pre-modern State had two overriding interests in marriage, and both were economic at bottom. The first was defining inheritances. In a society where everything, both status and property, depends upon legal succession through inheritance, defining who one's heirs are and what their rights are is of prime importance. This is also why laws concerning adultery and bastardy were so important.
The other interest of the pre-modern State was preventing poverty. Children brought up in stable households are more likely to prosper. The State will not be burdened by them; indeed, their prosperity will strengthen the State. Also, the more children you have, the more help they can provide the family with their labor, and the more likely that someone will be around to take care of you in your old age. So keeping marriages together in order to raise self-sufficient children who will support themselves and their aged parents was good social policy.
The Church had two overriding interests in marriage that brought them into collusion with the State, and both were spiritual at bottom. The first was promoting chastity. Chastity is more than avoiding fornication and adultery. Chastity can be either married or single. It can be vocational (as a religious calling), situational, or preferential. Encouraging a high view of marriage was seen as a way to encourage a spiritual virtue.
The other interest was in spiritual instruction. Believing wives and husbands would bring the other into the faith; meanwhile, choosing spouses among the already converted would double down on the spiritual influences in the family. Children of believing parents are likely to be raised in the Faith, and having the whole family come to church, participate in family prayers, etc., raises the odds of the young adopting the beliefs and values of their elders.
Leaving aside the Church's interests and speaking only of the State's, we see that the modern State has already jettisoned the concerns it used to have about marriage. Marriage is no longer concerned with property and inheritance as it once was. Nor does the State worry about keeping families together for the sake of the children, since there exists a large machinery of foster care and family law to see after their interests. And yet, the no-fault, no-penalty divorce industry and the blended families that are so common are not really in anybody's interest. They provide convenience to adults, but the children of such failed marriages are shown to be at risk of poverty and crime by study after study. One could argue that the State has not been well served by allowing its own higher standards of marriage to become so eroded.
I have said on several occasions that the problem with gay marriage is not about homosexuality, per se. It's about the devaluation of marriage for everybody. This process has been going on for a long time, and it can be demonstrated to be harmful in many ways. To continue to devalue it in order to fit gay people into it exacerbates an already bad situation. And by the time you've redefined marriage to include gay relationships, what's left is not even going to be of much interest to gay people. No, the State has a compelling interest in trying to raise standards of marriage, quite apart from the ideological interference of the Church.
The Church's problem is that we have had to compromise so much, that even we hardly believe in what we offer any more. But that's our problem, no matter what the State does.