Ick. That can't be right. I mean, even animals cut the placenta off right away! And as a person who was carrying around a baby kitty and it's placenta a week ago I can tell you that it's messy and cold and well, unpleasant really.
This is not categorically true with animals. The closest species to humans (orangutans, gorillas, chimps) do not 'cut' (there's that chimp with a cord clamp!) the placenta off always. Their primates have very long umbilical cords, which allows for nursing whilst the placenta is still attached. Particularly in chimps, placentas have been observed days after birth, still attached and simply are left behind after the cord detaches from the umbilicus. Placenta consumption is irregular, but does happen. Placenta qualities in primates are often identical to human placentas.
One of the best ways to cease post partum hemmorage in humans is to have the mother consume a tiny bit (like the size of a pencil rubber) of placenta.
I usually stay out of these discussions, with the hyperbole of disgust and whatnot. Guess we know who the non-medical types are!
But as expat_in_scotland pointed out, there are many critical health reasons to delay cord cutting, even beyond a few minutes. One of the quickest ways to get newborn anemia and maternal hemmorage, as well as risk of uterine prolapse (from manual removal of placenta) is to cut the cord upon birth. And heck, if the cord blood is that dandy to bank and save other people, the newborn has a birthright to that blood.
Yes, in my births we let the cord detach on its own. No, the placenta wasn't gross, messy, cold, unpleasant. This is the organ who fed my baby in utero for many months, I'm not supportive of tossing it into an incinertor or using to make cosmetics, shampoos.
It was seriously not a big deal to wrap up the placenta in a cloth nappy, after examining it for intactness.