You have stated in the above sentence that she was not following the law. That's not true. She just wasn't following the law in the manner that the lesbian pair expected.
The law requires the office of the clerk to process the licenses. It does NOT specify an exact manner or time frame, and therefore a standard policy that requires an appointment with an individual on the clerk's office staff is a reasonable approach to satisfaction of the actual requirements of the law - even if it isn't the expectation of the applicants.
That is how the existence of this standard office policy is related to your original post.
I understand your point. It’s no different than a municipality listing the time when a dog license can be procured or a recycling center being open.
It’s also a newsrag inserting an editorial statement, the town clerk’s religious beliefs, into a story about a marriage license. Again, an unnecessary piece of information.
Finally, would the marriage license have been granted if the couple were heterosexual? That information is not provided either.
My comment was regarding one aspect of the story, the clerk summoning a deputy to fulfill a task which was required by her position. You addressed a different issue, unrelated to my original response. If said couple were to appear at the proper time to acquire the license, one should presume a reasonable time frame based upon the laws of that state.