A lamb's umbilical cord separates as the lamb is born and either falls to the ground, or the mother gets up. As it separates, the cord is drawn thinly, which almost seals it, closing the blood vessels. The cord dries up quite quickly with iodine so reduces the likelihood of bugs getting in.
With human babies, the cord is much thicker and more substantial, and as far as I know, doesn't come apart when pulled. So to stop it bleeding or letting in infection we need to tie a knot in it, or clamp it. Even then, the human cord takes longer than a lamb's to dry up.
Maybe because a lamb is up and walking straight away, the chance of infection getting in to a thick, unbroken cord is too great, so natural selection has made the cord easy to break and seal. A human on the other hand takes about a year to start walking, and at least 6 months to start crawling, so a thick, strong cord is more important to the baby's survival, with an intelligent mother to carry it around, without the cord trailing on the floor.
That's my take on it. Pure speculation of course