I hate to interject in certain threads....this is one of them
BUT as a CP with sole legal and physical custody of one of my children: The NCP absolutely CAN take the child to the Dr. Certainly in an Emergency situation, but also in a situation where the child is in the 'care of' the NCP. ie; is visiting with the NCP and is ill and the NCP thinks they require treatment. ie; if they are coughing and have possibly bronchitis...or is sick and NCP does not know why, but wants to make sure they get medication if they need it. In fact the NCP could be liable if the child <for example> had a fever and did nothing about it, and the fever resulted in a more complicated physical condition due to the fever not being treated. (which can happen with certain high fevers).
I have learned this from both experience and observation. Not from trying to keep NCP from treating our child, but for example, once NCP called and asked about taking our child to the Dr. I said "OK" and then called attorney to find out how we could get 'proper authorization' for NCP to treat the child if/when a situation arose in the future, and was told all I have just said.
The issue with legal custody comes into play moreso in situations of consent for medical
procedures like surgery (or voluntary testing etc.), Psychiatric treatment/consent, major educational decisions and Religious decisions. If you go before a judge to complain an NCP took your child to the Dr. when they were sick, you'd look like a fool.
In an Emergency situation- Anyone can seek treatment, a neighbor, a teacher, a stranger on the street who happens upon an accident. It all boils down <according to the legal field> to what serves the best interest of the child and the childs Health & Welfare at that TIME.
My 2c from experience in my own life, and from observations in family matters' issues related to children's safety and wellbeing.
I am still wondering what the problem OP has with the other parent taking the child to the Dr.????