Even Davis was ultimately allowed to refuse to issue licenses personally; she just couldn't forbid the deputy clerks from doing it (only one of whom, her son, also refused). She also had an elected position with an oath of office, not an employment contract.

Actually, the judge ordered that the deputy clerks must issue the marriage licenses(save for her son), unless they wanted to be held in contempt of court.

I'm not sure how the path matters much. It contains language along the lines of "I will faithfully execute, to the best of my ability, the office of...."

I'm imagining having religious qualms about certain behaviors would fall under "best of my abilities"