Andy -- I think your explanation is probably right. It applies especially to the afterbirth proper.
Most mammals eat the afterbirth if they get the chance. As well as discouraging predators, it is a big lump of useful nutrients, at a time when there's a lot of repair and milk production to be done.
For den-living animals such as wild cats and dogs, the afterbirths and cords would become most unhygenic after a day or two...
The cord of calves snaps of its own "ac-cord", but the mother will often still nibble it off shorter. I suspect this is mainly to keep it out of the way, and to help prevent infection. Sometimes it's left quite long, and then seems to take quite a while to drop off. Within a few hours it shrivels up and becomes stiff and wire-like.
Cows don't usually eat the calf's droppings, except when cleaning their baby (curiously, calf droppings smell vile, but improve when they go onto solid food -- the reverse to dogs and humans).