I think many people would agree that one branch of economy where female labour appears indispensable, is health care, and especially the care of female patients. So how could this issue be resolved in a traditional society?
When we look to the past, most hospitals here in Europe were run by religious organisations, especially the Catholic Church (but not only). Nurses and sometimes doctors were nuns, the same is true about elderly care. In Lutheran cultures, as far as I know, there were also religious women who would perform this kind of job, the difference being that after some time they could quit and get married, unlike nuns (I read somewhere they had to serve for 5 years or something).
Further on, in some traditional cultures all the women got basic medical training, including midwifery. Birth only became highly medicalised in the 2nd half of the 19th century. Here in the Netherlands, many women still have the child at home with a trained midwife, and I heard it's not unusual in UK, too.
In fact, I once read a story on a British site which happened during the initial stage of corona, when a woman who was about to give birth phoned the doctor's office and they told her they couldn't sent anyone due to the pandemic but would give her instructions on the phone ( that was her 2nd child so luckily she had some experience).
I have a Dutch 1st aid book at home which has a chapter on what to do in the case some woman goes into labour and there is no medical help available, so even lay midwives don't sound so far-fetched.
In other countries, I heard, due to the shortage of medical personnel they ask family to come and spend nights in the hospital nursing their family. Here they just kick you out as soon as possible, and though we have a system of home care, they do ask nowadays if the relatives can perform such tasks as giving injections or wound cleaning (happened with my mother-in-law).
Of course, some types of medical care are very high-tech and/or need years of study and specialisation and couldn't be outsourced but many simple things certainly could which would further reduce the need for professional female work outside home.
Just a couple of thoughts...