There is probably no system that universally fits all circumstances. I watch with admiration how the Chester herd helps a new mom through pregnancy and throughout the reading of the calf. However, there are very few multi-generational herds like this in the US, and many first-time mom's have never seen a birth and don't always have the instincts to stimulate the calf to breathe or even remove the amniotic sac from its head. That's why so many zoos here would prefer to have pregnant mom's shackled and comfortable with vets in their space even before delivery, such as for rectal ultrasounds which are admittedly harder to do PC. Then too, calf birth and survival here are so much lower that human nervousness makes them want to be able to "control" the outcome more. In the 1993 birth of Kumai at NZP, first-time mom Shanthi had never seen a birth or a calf and, thinking that the newborn was some sudden danger, decided to step hard on the newborn's head. Eager, worried staff are anxious, if nothing else, to make sure a newborn is not stepped on.
The easy situations are the clearest. Columbus Zoo's Phoebe had many staff in her indoor enclosure during her pregnancy while she was shackled with one front and back leg. However, this isn't really an overall testament to its safety, because Phoebe was handled FC for many years and is comfortable with humans being in her space.
At the opposite extreme, there is really no advantage to handling a bull free contact. It keeps humans safer and eliminates any need for force for compliance, making a better animal/human relationship all around. Here, there no medical procedures for which sedation can't keep humans performing medical treatment safe.
I happen to be one of the biggest fans of Canada's American Lion Safari's huge (18-strong) herd of Asian elephants and their incredible multi-generational breeding success. This institution could probably not exist in the US, because all but the breeding male are FC, but that FC allows for magnificent experiences like meeting a neutered male after a performance and seeing 10-14 parade through the park from their enclosures to the huge lake twice a day, where one can witness moms and their daughters swimming and teaching youngsters to swim. I love this place and would hate not to have the chance to experience it, but a few months ago, an elephant giving rides unexpectedly attacked its keeper. I selfishly want to continue to see what is the closest thing to a real herd North America has and their behaviors, but even in this expertly-run place, there can be accidents.
Overall, PC has been a good thing for AZA institutions. There is no longer the need for force or threats to maintain a safe environment, creating a much happier relationship between animals and keepers. I suspect, however, it will take institutions here quite a while to trust in mom's delivering without medical and keeper presence to guard calves. Between dearth of breeding-age females, females who have not experienced a birth even by witnessing one, and crippling loss of calves to EEHV, calves are simply too precious here for everyone involved not to be nervous.