IIRC, CNN has something materially wrong in their article when they state that the couples asked for her to perform the civil service. I don't believe that is correct, they were attempting to get a civil marriage licenses after the Obergafel SCOTUS case decision, after the Governor had sent a letter explaining the decision to the clerks, and after consulting with the county attorney (who said she couldn't deny licenses to same-sex couples). The couples made repeated attempts between July and September to obtain a license and were denied either in person by Ms. Davis or by her staff under the direct orders of Ms. Davis. If that is incorrect, someone can post an earlier court ruling that indicates that, I would appreciate it. But this ruling notes that it was about not issuing the licenses.
I don't have an issue with a religious accommodation such that she didn't have to be the one issuing the civil license or performing the civil service as long as other staff were available, and that is the case. Where she screwed the pooch is at the point she ordered her staff not to do their jobs because of her religious beliefs.
It became not about her beliefs and actions and more about denying the two couples in question access to civil marriage licenses.