The process then, as I understand you to be saying,
a. apply/appear for license
b. paperwork processed by county
c. marriage solemnized/finalized
d. paperwork finalized.
I’m not trying to be a legalist, but WHEN is the marriage actually performed? For me, it’s step ‘c’. Whether a justice of the peace there at the county offices or by a solemnizer. Without that step there is no marriage on record.
However, I can see some Christians objecting to being a part of steps b through d in any way.
Better to have no one in the county office involved in this in any way other than if there is a contract between two people about inheritance, visitation, power of attorney, etc. and there’s some requirement to file this kind of contract as they do other kinds of contracts.
For example, can I contract with a bank to pay on a car without the county being involved in that contract?
State laws differ, but recording of a deed will almost always require involvement of the county or some other unit of government. Same for recording a marriage.
I am honestly not clear what I think about recording that marriage. An argument could be made that recording a second marriage of an adulterous spouse is also sinful. And if that’s the case, we have a much bigger problem than homosexual marriages.
The deputy clerk involved in the article I posted happens to be a conservative Anglican and has views of marriage which come from a theological tradition different from me, so I’m not comfortable explaining the details of her view. But for someone from a church owing its origins to Henry VIII, I can see why recording marriages involving serious sin short of homosexuality would not be as much of a concern as it would be to many conservative evangelicals or Roman Catholics.