A good clean shot through the vital area will drain the animal of most blood, especially when that shot is through the heart or lungs. This is true of the liver and most major arteries as well. The blood will be on the ground or in the body cavity.
Gutting does everything that sticking would do so far as "draining" the animal of blood after the kill. It is a good idea to hang the animal up as soon as you can after killing and gutting ... but having done it hundreds of times, I can testify that the amount of blood that drains after a clean, vital-area kill and proper field dressing is very small indeed.
I think cutting an animal's throat after killing in the field comes from the country practice of "sticking" animals when butchering. When you kill a hog or a steer (or whatever) you stun it to incapacitate it (we use a .22 bullet, properly applied), and then sever the carotid artery ... because the animal's heart is still beating, the blood is pumped from the body.
Trust me when I say that if you've never done this, it is a gory, messy scene. Hardly for the light of heart, or weak of stomach. But it's just something that has to be done, and is part of eating meat.
It is my opinion that there's no need to sever an already-dead animal's throat in the field. 