Just off the top of my head, I think if the County Clerk doesn’t issue a marriage license then you can’t get married in that county. Whether one can just go to the next county over or not I don’t know.
But, yes, it is all about the media and lawsuits and politics.
A marriage license is a state document. County clerks issue them as a convenience but you can get a marriage license in one county and then go to any other county in the state and get married.