I also think neither the perk nor the skill in general seems to work at 100% for
other parties (or the general %chance for saving troops lives as "wounded" instead of killed in battle). When my family member was appointed as my surgeon, with 280 skill, we lost fewer troops in every fight (% wise) than when she was leading her own party later.
These were not auto-calculated battles. And they were big: army vs. army with over 1000 on either side.
She had similar troop make-ups, but my party
always had fewer casualties (after my character's skill got over 275 and grabbed the minister of health perk, of course). Technically at the time I was checking (we were fighting the same battles, joined in the same army when I was noticing this)
she had the higher skill level, my character was sitting right at 275 and she was at least 290 by then.