Unfortunately, Lusztig seems to miss the point about the current trend towards "natural unmedicated childbirth being the correct, feminist way of giving birth" - we're seeing a rise in midwives and birthing centers as the majority of births CAN occur without medical interventions. It's not about (avoiding/experiencing) pain, it's about the mother being present to bond with her baby, rather than strapped to an operating table, waiting to be stitched up after a 5pm C-section so the doctor could go home to have dinner with their family. If a filmmaker cannot fact check relevant statistics ["I don’t know what the statistics are, but there’s probably equal numbers of medical C-sections that save babies and save mothers"] then why are they making a documentary on childbirth? It's interesting to learn about the history, however attempting to use a mid 20th century point of view, when cigarettes came with physician endorsements, to obfuscate the discussion around whether birth is a "medical situation", is hardly scientific.
I highly recommend Ina May's Guide to Childbirth for information on the history of medicalization of childbirth, and obviously the Business of Being Born too.