I have one rather practical answer, which does not answer the question in the body, but possibly in the title.

I was very surprised to learn that a (small) hospital in my home town was doing their scheduling ("rostering" might be the correct term, but I don't know) by hand.

They were clever enough to ask for help just in time. Again surprisingly: informally via a mathematician they knew privately.  Thus, they now have a professional university team developing a plan that fits their new needs - minimal service, minimizing the probability that a large portion of their staff simultaneously drops out.

I guess that such a strategy might also make sense for other community services, who might not even be aware of this fact.

(Disclaimer: I am not helping with the implementation, but I was lucky to know someone who knew someone who was a professional in this field, who luckily agreed to do it.)