Imo, the term "license" is not an accurate description of the document you register. A license is "permission" but the document is not about permission. It is not about meddling. It is not about the government being in bed with the couple.
The marriage "license" is a legal agreement, a contract based on the idea of covenant. When legal contracts are drawn up and signed, and witnessed, they then have to be registered with the courts to be granted legal status. When a contract is signed and not registered, it can be contested in court. When it is registered, it can't be contested, and there needs to be another legal agreement (with the original signees or by the court on behalf of a non-responsive signee) to override the original legal contract.
These are marriage and divorce contracts, and because marriage and divorce involve legal issues, they need to be registered. Registering not only legitimizes the marriage (and subsequent divorce, if there is one), it also provides information for vital statistics and census. Whether we like it or not, the government is part of our lives. It is part of everyone's lives, even remote villages that are not in contact with mainstream... every culture has their "government" system with laws, etc. Law in the Americas happens to be complex, but that does not mean it should be removed from our lives. God himself has established the laws and has created each of us in the very place where we should be, under the law where he wants us to be born and live. Government is involved from our birth - their involvement with the hospital where we are born, the registration of our birth name, etc. A marriage "license" is no different than the "registration" of our personhood as an infant, in that it provides us with certain rights and certain responsibilities.
God is a God of covenant, or an even higher form of contract, one that is not as easily dissolved as a contract. God's ideal for marriage is that it is covenantal rather than just contractual. This is why we have a "solemnity" in our vows and in the signing of the contract. I don't think this solemnity is stressed enough in our culture or in our pre-marriage counselling. God himself has set a precedent for us in establishing the solemnity of covenant, when he walked the blood path in his unilateral covenant with Israel. The relationship between God and Israel, and with Christ Jesus and the Church is what marriage is supposed to represent...starting with this solemn covenant. Although God's covenant with Israel was unilateral (meaning if the covenant was broken by either party, that God alone was responsible for providing the payment for its brokenness....which is why he sent Jesus as they atonement for Israel's breaking of the covenant), the marriage covenant has always been bilateral, making each fully, 100% responsible for the covenant and repairing any brokenness in it. This is congruent with the NT where it talks about not making sacrifice until all your relationships are reconciled, and where it talks about mutuality. In the OT, it was the fathers of the bride and groom who walked the blood path, to demonstrate their faith in their children and to demonstrate the solemnity (severity) of the covenant. Iow, blood would be spilled by the family if the covenant was broken, just as in the covenant of God. It was also all negotiated, contractual, and witnessed by the community. I'm not sure when census started, but when it did, people had to register along with their spouse. Over the centuries, this has evolved to what it is today - registering your marriage with the government for statistics and legitimization.
We are to obey the laws of the land, and the laws of the land grant certain rights and responsibilities when people are legally married. As has been stated, there are practical reasons to marry with the "license" (actually registration). But there are also spiritual reasons to do it....to use marriage as a platform for living out their Christian testimony to the relationship between Christ and the church.
Those who choose to live together and try to justify their choice, will not have to justify it to us... instead, they will have to stand before God one day and account for why the chose not to reflect the spiritual union of God and Israel / Jesus and the Church. Then God in all his holiness will decide whether or not you are justified. Personally, I am not willing to take that chance, and registering my marriage with American and Canadian government is not contrary to God's Word, so I would rather err on the side of holiness.
There is also the matter of the woman at the well. Jesus asked her to go and get her husband. She said she had no husband, to which Jesus replied that she had had five different husbands, and the man she was currently living with was not a husband. So living together is not the same as marriage, no matter how you try to justify it.