Your story raises red flags with me, and I don't think it confirms the point you were trying to make, but its partially it's because we don't know your background and experience. Were you icu/ Er trauma trained at this point? Had you cared for vents/ post arrests before. If not, that was a very unsafe assignment to accept as you would have had a huge knowledge deficit.
I disagree that you shouldn't move a patient before the ett is confirmed, but every hospital I've worked at has a portable X-ray team. Had the X-ray been done already? Why couldn't the Er doc have confirmed the X-ray then and there. I would argue that delaying the patients arrival to icu is far less safe than waiting for cxr placement. At least in icu, you are surrounded by resources and supports for your patients: nurses who are familiar with the drugs needed, intensivists etc. My experience is that even if the intensivists have written orders down in ER, unless it prevents the patient from crashing, ER just doesn't have time to implement the interventions, including targeted temperature management, I would be concerned about delaying interventions such as these.
I agree that doctors will sometimes ask us to do things that are outside our scope of practice, or against policy. But often explaining what the policies are is enough. With this story, I don't think the resource nurse was outside her scope of practice to escort the patient. It as when she received an order for a medication she clearly wasn't familiar with and gave the wrong one and wasn't aware of the required monitoring afterwards that the nurse decided to practice beyond her scope and this event unfortunately happened. She made several errors including not doing the basic rights of medication.