The standard procedure for the SS is to take those they’re guarding directly to a hospital in a medical emergency. Yet they were overruled by a staff member (we don’t know who it was).
The staff knew what to do and they must have planned it ahead of time because there was no hesitation. They went straight to Chelsea’s place.
Which means they know exactly what is wrong with her (not pneumonia) and exactly how to treat it in a way that she revives quickly. If they didn’t really know, like if there were complications related to pneumonia, she should have gone to the emergency room.
I believe they have contingency plans set up wherever they go. That’s why she doesn’t do anything but very tightly controlled public appearances.
In Seattle the Gates family had backdoor privileges. I am sure Hillary does as well.