Human babies only have about a cup of blood *total*. That means that if there is any bleeding from the cord, their blood volume can be compromised quickly. The vessels at the cord stump are large, and if there is anything preventing them from being completely occluded, such as delayed transitioning or a blood clotting issue, then they might bleed out enough to be life-threatening, or less than ideal. The cord clamp is a fail safe. It also depends on how short the stump is left, the longer the stump, the less chance of bleeding. Also, a little bit of bleeding from the cord over time can add up to a significant volume. More than a few drops of blood is too much. The point of delayed cord clamping is to save blood volume, not clamping the cord could result in losing some of that precious blood. Keeping the placenta attached to the baby poses little or no risk of bleeding as it is a closed system, which is a great benefit of lotus birth.
In animals, the cord is crushed when bitten, allowing natural clotting processes to happen. Simply cutting the cord doesn't do this to the same extent. Also, animals seem to leave the cord relatively long, at least for a bit. If the cord is chewed too close to the belly, the baby animal is at a high risk for bleeding out.
If cord clamps seem too intrusive, there are cord bands. Dental floss can actually cut right through the cord when used tight enough to clamp the cord. The clamping device, and cutting device, should be sterile, as bacteria can enter through the cord until it is dry and hard.
One big component of why the cord is so big and juicy is the Wharton's Jelly surrounding the cord which diminishes rapidly after birth. The pale, floppy cord may still have vessels patent enough for blood flow for several hours after the birth: one can catheterize the umbilical cord for *hours* after the birth if the stump is left long enough if venous access is needed in emergency.
I would also venture a guess that the mortality rate for birthing animals and their little birthlings is much higher than we would accept for humans today.