I understand your frustration over the director calling you after you had indeed picked her up. And yet, I'm sure she was distracted by something before she made the call. I doubt it means she really thought your daughter was still in the school.
As for not being watched.. Do you really think providers and caregivers can have their eyes on one single child the entire time? Even a nanny of one child could not do that. My mother, my daughter, and myself are in this house with 5 children every single day. My 11 year old is doing her homework. But she loves the kids and is a great helper. My mother is 66 and isn't "in charge". But she keeps an eye and ear out and she helps. I am in charge and I must do things like sweep, pick up, prepare food, change diapers, talk to parents, answer the phone, put the laundry through, and the list is literally endless. Even with 3 people that are old enough and responsible enough to watch for behaviors like running, pushing, shoving, biting, etc.. Sometimes, stuff happens.
Now, you imagine having several rooms, one director, one or two teachers per room and at least 4-8 kids per provider depending on the ages? I'm not even sure of the numbers in New York. But there's no way that you can expect them to have eyes in all parts of their heads or to have full-time support staff to do nothing but clean up after everything they do. These are providers that are not necessarily doing any personal laundry or housekeeping as such, but they are preparing lessons, filling out daily sheets for parents, filling out accident reports, dealing with discipline, teaching, and you hope and expect they are cleaning and inspecting toys, etc.
I am very sorry about your child's scar. But that can happen anyplace.