How can it be made family friendly? Genuine question. I used to work backstage before children, too, but made the decision to work in another area before I had a child as I was well aware that it simply wasn't compatible because of the hours. DH still works in theatre (freelance) but quite honestly I can do other things and he can't (I appreciate that this isn't the case for everyone). Also, I was not prepared to give up seeing my child in the evenings/after school and putting her to bed every night. Thankfully, now that he's freelance, he often gets weekdays at home and is able to do the school run etc and spend time with her. It was very difficult when she was younger and he was still in the West End. She genuinely did not know him very well (and didn't like him much as a consequence).
While it could be possible to spend time with your child while it is still a baby and work in theatre, it would be the case once the child is school age that you simply would not see him or her (maybe not at all if doing Sunday shows). What would be the point of that? I don't see how it would be nice for any of you. It was one of the major factors in DH going freelance, which thankfully has worked out really well.
I suppose the only way to continue working in theatre is to find a job in a similar/related field with family friendly hours until your child is an adult, if it's really that important to you. I worked in Wardrobe and know quite a few mothers who became makers, dyers, milliners etc after children.
As you say, on eight shows a week, you cannot be involved in raising your child in a sufficiently hands on way. And even if you could have a nanny, would you really want your child never to be put to bed by you, to never see you after school, to never be able to spend an evening with you (bar once a week)? You wouldn't be able to help with homework, go to parents evenings, take holiday over Christmas (so would need childcare then) etc etc etc.
I suppose if there were more jobshares available, it would become more possible. But would that really be fair to the rest of the company? I know most performers wouldn't like for instance having two different dressers on different days. That kind of thing is a very personal relationship. Not sure about Stage Management but I can see it being a bit disruptive having two people sharing a show job. LX might be better for this than any other area bar FOH. Freelancing is probably the best option for your situation, depending on your field - this would probably require you to work a lot of weekends and you'd need either a supportive partner who is available at weekends or nearby family.
As for female doctors and nurses, none of them will be working exclusively in the evenings/weekends as there will be a shift pattern so they will get time at home when their children are awake as well as having to work sometimes at night or other anti-social hours. Plus part-time work is more viable. And doctors earn a lot more than theatre technicians so can presumably make more use of nannies etc. Nurses can go part-time and do family-compatible hours only, I presume, this keeping their skills up for a return to more hours in the future.
I don't think au pairs would work for you. I can't see most of them wanting a job where they can never go out in the evening. On a shift pattern like doctors or nurses, it might be more viable as they might only be doing nights every other week or something (sorry, not sure of details of medical shift patterns).
Re requesting flexible working - you can request but there is no requirement to grant it. And who would want an employee who was never ever available at the busiest times?