Unfortunately, the government must have a role in marriage, since child custody disputes must be adjudicated by the government. Try to imagine adjudicating such disputes without marriage records.
Moreover, marriage, i.e., a lifetime commitment between a man and woman for their mutual benefit, and the begetting and raising of children, is a natural institution. Families are the building blocks of society, and the State has a natural obliglation to regulate and foster them.
Many Christians mistakenly believe that marriage is a purely religious institution. It is not. But Christian marriage and natural marriage are different. Christian marriage can be sacramental, and its purpose is more noble than natural marriage. A Christian marriage is a lifetime commitment between a man and woman for the purpose of each attaining heaven, and the begetting and raising of children for the same purpose.
You don’t need marriage to ajudicate child custody issues.