I would suspect a terminal ill patient would be under the direct care of a health care provider or family member. If you are talking about a patient whose injuries result in a fatality prior to arriving at the hospital, the body likely becomes the property of the Medical Examiners Office (MEO) until a cause of death is determined. An individual who dies at the hospital may or may not be subject to autopsy by the MEO. In any case the hospital and the family/attending physician would be the “agent” to contact the family regarding organ donation, not EMS/Rescue providers.
The lawyers on the forum can correct me if I am wrong, but I believe in most if not all 50 states, upon death (non suspicious death) your body becomes the property of the family. Your wishes as to your body’s final fate are really not up to you, despite directives in your will, organ donor card, driver’s license directive, etc.
Pete