Agree totally, can't possibly be stable or very comfortable to shoot like that at all.
Back in 1979 my dad shot my birth with a 50mm lens on a Pentax K-1000. He made an album. I now possess the only known copy of this album. I am seriously considering burning it :)
Seriously, though - neither parent nor myself really wants to look at it. If the day ever comes, I highly doubt I would pay for this service based on my own experience. A nice portrait of the baby and the family at some point after the fact (and the cleanup), possibly.
I wasn't in a position to photograph anything too disturbing. He was cleaned up before the first image, which I think was after I cut the umbilical cord. We love those photos. They show that he had a personality from the moment he was born.
Since the maternity ward was too busy that night for my wife to get a private room, I couldn't spend the night there, so I went home and souped the film, scanned it when I woke up, and e-mailed images to the family before returning to the hospital in the morning.
I shot Tri-X pushed one stop with a fast lens, no flash, incidentally, and was surprised to see the photographer in the article using flash, even bounced. I think it's a myth that it could be harmful to the infant, but who wants the annoyance in such a situation? There's plenty of light in a delivery room.
My sons are now 5 and almost 7. For both, I filmed the birth on a digital video camera, including myself cutting the umbilical cord. I didn't include any gory detail in the edited films. Both boys love watching their arrival into the world. For both I also took a real film photo on my Hasselblad of the obstetrician holding them minutes after their birth.
I had the extreme fortune of taking probably the best photo I've ever taken a few moments after the doctors handed my wife our daughter. Nothing gory, just pure emotion. My wife has to work to not start crying whenever she sees it. I find that funny and hugely flattering.
When my son was being born I had a portable radio and some headphones with me.... BBC Radio Four was broadcasting The History of the Guitar in Jazz... I didn't want to miss that!