Translation:The skilled doctor heals many soldiers.
Physician is a better contemporary translation than a generic term, doctor. It should read medical doctor, to be precise. The word doctor can be used, nowadays, to describe anyone who had earned a PhD title. From the context, of course, we can surmise that it is not a Doctor of Jurisprudence who does the healing.
It is definitely not true in British English that 'physician' would be a better word choice. 'Doctor' is very widely used to mean medical doctor, 'physician' is used much more rarely by laymen, and 'doctor' on its own is essentially never used to mean someone with a PhD, unless by people from outside academia who guess wrongly about what the convention might be.
I guess the situation in American English is fairly similar, but hopefully an American can weigh in.
You might have a personal preference for hyper-precise use of language, but it is definitely not the case that everyone else "should" go round saying "medical doctor" instead of "doctor" every single time.