I don't think it's appropriate to take a charge to the dentist or to the doctor, if you are going to need an examination. In those situations, you are unable to look after them properly. I should imagine most SAHP arrange some kind of help when they need to go; a friend or a family member. I suppose if a nanny took a friend along, it would be okay, so long as the DC were properly supervised.
If a nanny had some errands to do that could be incorporated into the day, I would be happy for them to do them. So, if they wanted to return an item to a shop, and tacked it on to a trip to a soft play venue, that would be fine.
Also, if they had to wait in for a repair person, once in a while, I would be okay with them taking the child to their home. But I would expect to be asked. I wouldn't be happy with a nanny going to their own home every day. If I wanted that, I would have a child minder.
My nannies would sometimes visit their parents, or other family members, and my DC benefitted from this. One nanny had a sister with a baby, so it was a lovely experience for them to visit her, every now and then.
I see that your friend works quite long hours on her working days, but she has three days off a week, one of which is on a week day. She will also be entitled to about 23 days of annual leave. I'm struggling to feel sorry for her, frankly!