Consent would connote she personally is condoning the gay marriage, when in fact she is not, but it is the county/state for whom she was elected. Nowhere in the legislation does it convey the authority is vested in the actual person, it conveys it is vested in the office, which if filled by a duly elected official.
Extract from the Kentucky law.
Since the time when the District Court replaced these courts, the clerk has no longer had judicial duties, and the name of the office has been
abbreviated to county clerk to more accurately reflect the nature of the office.
The duties of the county clerk fall into the general categories of clerical duties of the fiscal court: issuing and registering, recording and keeping various legal records, registering and purging voter rolls, conducting election duties, and conducting tax duties.
Nothing in the above gives her ANY authority to refuse legal applications.
Her belief system does NOT enter into it, both morally and legally.