The biggest issue was that people didn't like government services closed on Fridays. But instead of doing shifting or trying anything to mitigate that, they simply gave up.
The depressing thing is that this pattern happens over and over, be it remote work, work hours, work days, work dress, or a million other things. Society just "pins" normality arbitrarily then keeps returning to that.
[0] https://www.npr.org/templates/story/story.php?storyId=102938...
We need to automate a lot more of the public and financial sectors than we currently do. As long as a significant portion of work needs to be done face-to-face, there will be pressure to synchronize everyone's working days and hours.
In order to utilize shifting in a typical government office, you will need to train a lot of people to perform many different tasks involving many different laws, and come up with new procedures to assign clear chains of responsibility. Especially if the shifts are as complex as you describe. Utah might have decided that it's not worth the hassle.
Besides, if you could simplify and streamline a task so much that any on-duty employee can handle it, you might as well automate it all the way. In other words, tasks that are well suited for shifting also tend to be well suited for automation. The only exceptions are 24/7 jobs such as the police and emergency services.
I agree that shifts are often not possible.
A flexible four day workweek would help a lot with this. Got Tuesdays off? Find a dentist that's open then.