Given your usual thoroughness, I am surprised that you did not mention Ignaz Philipp Semmelweis, a doctor who went mad trying to convince his stubborn colleagues to wash their hands in transit from autopsy tables to birthing beds. (This idea did catch on a bit later).
For decades I’ve known both the “gives the family something to do” story and the “doctors need to wash their hands” story.
What I only realised late in life was that doctors were learning advanced obstetrics, in particular manipulation of the position of the baby, by practicing on corpses. Particularly, on the corpses of women who had died in childbirth, or just before, or just after childbirth.
This was a major medical advance. Breach-presentation often caused the death of both mother and child, but, as you’ll know if you’ve watched any veterinarian on TV, you can save both mother and child by reaching in and turning the baby around. (With a cow, you might put your whole arm in. With a human, you might use a steel obstetrics instrument.)
In America now, you’d go directly to cesarean if you had a difficult presentation, but pre-anasthetics, pre-sterilisation, surgery was not an option.
The first educational method was using dummies and models, and there was big medical dispute between “old school” and “new school” teachers of these radical new obstetrics techniques, but training on actual bodies was clearly better, and probably cheaper.
So in the advanced, cutting-edge medical schools, obstetrics trainees were literally going from the autopsy theatre to the birthing suite, and the only people who thought that was a bad idea were political opponents and the technically inadequate.
I get it now. The water is to wash their hands before the birthing.