Native English, yes.
It is an English idiom, but it goes back to times before electronic communications existed. If someone needed a doctor, then someone had to go and physically find a doctor. Meanwhile whoever did the sending would stay and try to tend for the patient to the best of their ability. This was universal.
Esperanto came along AFTER the invention of the telegraph, and almost contemporaneous with the first telephones. So it would be surprising, to me at least, to hear an esperantist say Mi devas sendi por kuraciston, especially since calling for medical help is so easy all over the world. I've heard stories of breechcloth wearing New Guinean islanders carrying iPhones and spears. If some of the most primitive places in the world have telephones then calling (voki) a doctor is more likely anymore. This is the new universal.
Telefoni says that one used one's telephone, of whatever sort. Voki just means that someone was called for by whatever means were available, which can and does include telephones, radios, smoke signals, semaphore, sending younger siblings, or simply screaming at the top of one's lungs.