I am not sure that no-one IS irreplaceable, Postchildren. I worked as an operating theatre nurse, and if I was the only qualified scrub nurse in that theatre (something that happened every day), it could be very difficult indeed to find another suitably qualified nurse to take over from me, if I was urgently needed to pick up a sick child.
There are very real safety issues involved - policy when I was working was that the scrub nurse who started a case should see it through to the end, unless there was a really good reason why she should scrub out (ie. she became ill, the case went on for a number of hours), because the scrub nurse knows what instruments they have on their trolley, including any extras, and even though a count of the swabs and sharps was done at a handover, mistakes were possible - and though this was unlikely, it was not a risk that would be taken lightly.
And that's just the scrub nurse - what about the surgeon? They can't step away from the table mid-operation, unless there is another surgeon, who knows the procedure, available to take over. Even if you wait until the end of the case, and then the surgeon leaves, what happens to the rest of the patients on the list. if no-one can take over? They get their operation cancelled and are sent home.
There are plenty of jobs where you can't just step away at a moment's notice. And with staffing levels so low, in so many jobs, it can be well-nigh impossible for someone to leave.
There is no single easy answer to the sort of situation the OP describes. I have the greatest of sympathy for the poorly child, who just wanted the comfort of their own home. I have sympathy also for parents who have to work, to pay the bills and put food on the table, and who cannot easily leave work. It's not like the parents were in the middle of a nice day out and refused to come home and pick their poorly child up. I suspect they were between a rock and a hard place.