Samantha Wildman was a science officer, yes. Doesn't necessarily mean she was a medical officer.
True, but a competent xenobiologist would be off to a better start than a half-trained chemist with learning how to be a doctor. She'd likely have training in at least some of - botany (including plant diversity & physiology), bio-chemistry, biotechnology, classification (animals and plants), cytology, ecology, environmental science, genetics, molecular biology, physics and zoology - if not all. Whereas Paris as... chemistry. Huh? (He could be the dispensing technician - ala Tarses, but nurse?).
Essentially, the EMH as designed is supposed to be a nurse, not a doctor. Pressed to work all on his own, he makes for a competent doctor, and amusingly assumes a bedside manner fitting of a self-important specialist - but said bedside manner is also natural for a laborer who's not supposed to do any bedside to begin with, merely to hand laser scalpels and sonic wipes to the actual practitioners of the healing arts.
Sorry, got to disagree, a lack of beside manner indicates that if anything he's intended to be the surgeon, as they mostly deal with unconsciousness/dying patients, beside manner is not emphased in their training. On the other hand, traditionally patient moralle and welfare was arguably the primary job of the nurse (a role that as drifted down to the minimally-trained HCA and ancillary staff in recent years.