If a couple has all necessary powers of attorney, wills, and cohabitation agreements in place, the only things a "civil marriage" can offer is the ability to file taxes jointly as married for tax benefits, the right to Social Security benefits upon a spouse's death, and a free name change.
One thing that a civil marriage does is it binds you legally in a manner that should the marriage fail, you have to go before the civil courts to divorce. Paul admonished against going before unbelieving courts:
1 Corinthians 6:1-8 King James Version (KJV)
1 Dare any of you, having a matter against another, go to law before the unjust, and not before the saints?
2 Do ye not know that the saints shall judge the world? and if the world shall be judged by you, are ye unworthy to judge the smallest matters?
3 Know ye not that we shall judge angels? how much more things that pertain to this life?
4 If then ye have judgments of things pertaining to this life, set them to judge who are least esteemed in the church.
5 I speak to your shame. Is it so, that there is not a wise man among you? no, not one that shall be able to judge between his brethren?
6 But brother goeth to law with brother, and that before the unbelievers.
7 Now therefore there is utterly a fault among you, because ye go to law one with another. Why do ye not rather take wrong? why do ye not rather suffer yourselves to be defrauded?
8 Nay, ye do wrong, and defraud, and that your brethren.
Paul said that it would be better to suffer being defrauded, or ripped off, than to go before the unbelieving courts of this world to settle our disputes. In fact, for old world groups like the Quakers, refusal to be subject to civil authority is considered a part of one's Christian testimony.
Ultimately, I think it is up to individual couples to decide what works for them. The civil marriage system is there to participate in if a couple wishes. But for many couples it would be far more of a liability than a benefit. If these couples enter a covenant commitment and wear wedding bands to publically affirm that they are in union with another, I don't judge.