We ran across this once when I was nanny. She is a Masters Degree level professor of nursing at a local college. Every doc and nurse in town know her and her kids.
She called the pediatrician and the local hospital. They both said they would not accept a letter that was brought in no matter if they knew both parties. Too many people are computer smart and can make forgeries. They required a consent form to be signed in their offices. She went in to the pediatrician's office, filled out their form, showed her drivers license, signed it, the office manager signed it, etc...it was allowed and had an expiration date on it.
The hospital had a similar form. They had her show her license, fill it out and sign it, and then had it notarized by their staff. That way they knew the paper was legitimate. They put a note on the kids names where if their names came up they would see the flag and the copy of the paper was shown. One of my friends girls stepped on a tile brick around our trees/flower beds and sliced their foot open. Because of the forms we filled out in advance at the hospital they only had to put her name in and I was cleared to sign requesting treatment.
Even in child care or school field trips, if a child is hurt and requires emergency treatment the ER calls a judge to sign off, even with a form. It can take time too so don't think just because you have a sheet of paper it solves everything.
There is a type of limited guardianship too. It allows only certain things, such a seeking medical treatment. The law is only a few years old and not known well. It is a court paper and they are never disputed. It can have an expiration date on it too. It is a step above the paper saying they have permission to treat. It is not that specific and can come in handy when the unknown may happen.