Weird, when you google "Kaci Hickox" "private plane" a bunch of stories come up, so I am not sure what really happened.
In working back through the reporting, she agreed to go to Maine "but not via mass transportation." I'm led to believe that the NYC-area media, where no one travels by car and where Maine is seen as a million miles away, translated that mentally and wrote "she plans to travel by private plane rather than public plane" without understanding that for most of us, a car trip is more the norm than a plane and Maine isn't too far for a car trip. She's not a NYC girl by any means and wouldn't think like one. JMO.
We had this note in the Wash Times: "A private carrier will take Kaci Hickox back to her home in Maine, the Republican governors administration said." "Private carrier" (if that was the admin's wording, and not the article writer's) can be interpreted a number of ways, and interestingly, that same article's headline writer decided it would be by AMBULANCE.