I think the problem is that there is a false hierarchy of roles clouding the issue. In our 62 years of living together, we have developed roles based on preference, biological difference and skills. Madame handles the financial affairs because she was once a bookkeeper whereas I am bored by bean counting. I chop the logs because I am bigger, stronger and more skilled with an axe, although the small house axe I use is the one she used as a girl before she met me. I do the cooking because I like doing it. I look after the vegetable patch, she does the flower garden. We had no choice as to who gave birth to our two sons, and she was much better at the nitty gritty of child rearing, and I went out work for the money to live on while she child-minded at home in the early years. I would suggest that all these roles are of equal importance and therefore worthy of equal respect.
Hugh and Fiona's current situation highlights another important issue; roles can change dramatically and unexpectedly because of circumstances. Flexibility and the ability and willingness to carry out a different role becomes paramount. In our own case, being in our 80s and consequent health issues has meant adjustment to our roles. For example, Madame can no longer drive, so I do the shopping.
Similarly, in the wider world, some jobs are seen as more important than others and gender roles ascribed to them muddy the waters still further. Refuse collectors, doctors, entertainers, engineers, farmers, teachers, administrators and entrepreneurs all have their roles in keeping the wheels of society turning. Who is to say which are more important than others? I think there is a case for saying that some roles are better suited to men and some to women, although I would not agree that anyone should be prevented from exercising their talents or preferences on the grounds of gender.
In my view, there is a confusion between sameness and equality. Men and women are not the same but they should be seen as equal and worthy of equal respect. Individuals or societies which regard men as superior to women are plainly wrong.
PS Get well soon, Fiona.