I disagree. The triage nurse was the one who screwed up in your example.
The myth that you get admitted sooner coming by ambulance over walking in is just that, a myth.
I have been on quite a few calls where this was explained to the patient before transport and they ended up getting a $900 taxi ride in an ambulance only to be sent out to the waiting room.
They don’t have a triage nurse if you drive in, or at least not in the South, where I live. They have some women with long fingernails and lots of extensions sitting behind the desk watching the soaps and asking if the patient you have brought in can stand up.
Believe me, going into the emergency room in an ambulance is the only way to go.