Not inherently very different from a blood transfusion.
If you don't kill the baby to get the cells. Otherwise-discarded umbilical cords pose no huge ethical dilemma.
Can stem cells from umbilical cords be cloned?
If so, another reason to save cord blood.
I used to work as a surgical technician in a hospital delivery room. The amount of blood that comes from the cord is astonishing -- it is very dark and looks very "rich," I don't know how else to describe it. Much darker than the blood you get when you cut yourself.
When I worked in the OR, the doctors would clamp the cord at the umbilicus, and use another clamp at the placenta, and then cut the cord and let the blood drain into a stainless steel bowl on the floor. When the placenta was delivered, they let that drop into a bowl, too, and examined it to make sure it was intact.
I dimly remember sending placentas to pathology but can't remember if the pathologists examined every one.
The blood was poured down a sink.