She's getting paid to entirely offload one of her duties to someone else. Beyond the obvious discrimination motivation, it just plain makes her bad at her job.
I would assume that actually signing marriage licenses is a relatively small part of what a county clerk does and almost always delegated to deputy clerks in practical terms, and SSM will account for certainly fewer than 10% of all marriage licenses to begin with. The reason she got thrown in jail wasn't because she wouldn't issue same sex marriage licenses, it's because she tried to stop her office from issuing marriage licenses.
Also, since she doesn't have a boss and she is the boss of the people she would be delegating to... like... uh... does it make you mad that any manager delegates any work they'd rather not do to subordinates? That seems kinda a routine part of business.
Note my only experience with marriage licenses was getting married, but what I did was I went to one of 8 offices in my city that issued marriage licenses and got served by one of about 15 wickets at the office, and the process took I would say about 3 minutes. So I don't think if some particular person there had refused to serve me on whatever grounds, everyone else would feel like they really got a ton of work dumped on them.
(All of this is proceeding from the exchange you had with Cyan, so in this hypothetical she is not issuing licenses but also allowing her deputies to process them and no one seeking a license is being turned away--which is what your post was about. If she doesn't allow her deputies to process it, obviously she's still in contempt)