There are countless instances where employees are allowed adjustments to not go against their consciences. I grew up in an era where all males were subject to the military draft, but one could apply for conscientious objector status to avoid military service. However, there was an expectation that in being granted that status, the individual would serve in some other capacity. The difference between Davis and an employee is that Davis is an elected official and has taken an oath to uphold the law. Employees usually do not take such an oath. That is a pretty big difference. If Davis were an employee and didn't want to be involved with gay couples due to her beliefs, I'm sure someone else in the office could assume that duty. Unfortunately for her, as an elected official sworn to uphold the Constitution, Davis doesn't have that luxury. There were likely a bunch of officials throughout the country whose worldview was jostled by the Civil Rights Movement. If they were elected officials, they had the same choice then that Davis does now.
The state is fallible by its very nature. It is man-made. It cannot meet the needs and desires of all. Hence, a consensus needs to be reached and, again, by its nature, consensus will not meet with 100% of the desires of 100% of the people.