I don't think it should be on them if the school is in a bind. That's a lot of pressure for someone that also has a life. If things happen, they shouldn't feel guilty for prioritizing themselves first. I also say this as some who will be leaving my school before the school year is over. I plan to make the transition for whoever replaces me as easy as possible but it's not my burden to bear that they may not have a permanent nurse once I leave. I think the city should be better prepared and offer school nurses more money so that the position is attractive to applicants.
I've been a school nurse now for a little over 4 years in an elementary school. It's not clique-y like other jobs with other nurses but it's like working alongside people who think this is high school and a popularity contest. If you're new to schools you'll soon find out that this is how adults who work in schools behave. It's a sad reality. The most difficult thing for me was realizing that THIS was true autonomy. I could call my supervisor, sure. But nursing judgement takes precedent and you have to make the right call. It's possible that you will make mistakes along the way but it's better to realize that mistakes sooner rather than later before it becomes a sentinel incident. I was very into my policy and procedure manual so I would call 911 if there was every anything that I felt I couldn't handle and was very emergent. Better safe than sorry and covering your @ss comes into play a lot. You're essentially on your own, in charge of those below you in your medical office and THE medical representative in that school.
Best of luck with whatever you decide to do.