common folk are cured as much by the magic as by the doctor
Can't let that go unanswered.
What you suggest is only true sometimes.
When we speak with Latin words, or words that come from earlier times (like "gallop rhythm") the reason is that they have a very precise meaning, not always the same as the Standard English meaning. They also come with very specific implications about what is happening in the body and what might have caused it and what interventions might make a difference one way or the other.
Would you rather the "gallop" had been called by its technical name? There are two kinds, actually, an "S3" and an "S4," meaning a third heart sound and a fourth heart sound, but that doesn't really help very much. (As a specialist I often think half my job is to translate from Latin into English!) Neither one means anything like "the heart beating so fast the patients almost inevitably die of it," and they aren't always "complications of heart surgery," and there _are_ several treatments for one of them, and the other requires no treatment anyway! But that's all compressed into one or two words.
Every field has this phenomenon; it's called jargon. It can be expanded when desired - as when medically sophisticated patients want to know more about what's happening to them and why and what to do about it - but between doctors it's much more efficient the short way.
Physicians don't do magic, and we know it. Maybe 85 percent of sick patients get better just with the passage of time; if they happen to see a doctor just before they improve we're happy to take the credit, but we know full well that the patients would have likely have gotten better anyway. Or if they had seen a chiropractor. Or a grandmother. Or a shaman. Or a voodoo specialist. Or just stayed home in bed. Or even ignored it entirely and gone about their usual routines.
And another ten percent of patients are _not_ going to better, not with the best doctor in the world, or any of those other kinds of healers either.
But then there is the five percent for whom we do make a difference, and that _can_ be shown by science. Those are the ones - not the slightly sick, not the worried well - for whom it really matters what kind of practitioner you see. For the others, we provide reassurance that it's not anything more serious. And yes, many others could do that just as well, and maybe in some cases better.
As long as you're sure you're not one of the needful five percent.