"It's obvious that JA expected her readers to agree with saintly Anne that nursing is a woman's natural task, "

This is not obvious to me. Anne really doesn't want to go to the party. Here she has a convenient out to have a quiet night to herself without having to face Wentworth with a welcome distraction in the form of caring for the boy. All she has to do is say what the Musgroves want to hear. Charles wants to hear that it's a woman's place and Mary wants to have permission to leave because she suffers from "excessive feeling." (Yeah...for HERSELF). Anne says what she does in order to get rid of them, I think. I don't necessarily think that is entirely saintly, nor evidence of JA agreeing with the idea that nursing is solely the province of women, but rather specifically Anne is useful and practical and Charles and Mary are useless and impractical in a sick room.

If it were really the case that JA bought into the drivel that it was not a man's place to be in the sickroom, why is Edmund who saves Tom's life in MP and not one of his sisters? The answer is that his sister's don't really care enough to interupt their holiday. They are just as useless and callous as the Musgroves.

Even if the men weren't doing the nursing themselves, they were still expected to be there. Think of Col. Brandon in S&S and Bingley in P&P. While they are not actually by the side of their beloved, they are waiting in the wings in case they are needed-- fetching doctors, family members and providing moral support to the women doing the work.