I have been thinking about letting a deer bed down, and I don't think it is such a great idea. What if the bullet hit it but you let it bed down and it heals? Why not let it run and keep the wound open and bleeding?
The vast majority of the time there would be no benefit to immediately pursuing a wounded deer. The only instance I can think of would be if you happened to nick a large artery or major vein. Then perhaps pushing the animal would prevent a clot from forming and keep it bleeding. It's been my experience that this would be pretty rare and you would be able to tell because the animal should bleed rather profusely at least at first. The only time I would push a wounded deer is if I had blown one of his legs off or hit him in the neck. In other words a shot that I KNEW had not possible hit anything besides veins or arteries. A shot through the abdominal or thorasic cavity on the other hand and you would be well advised to wait. The vast majority of wounded animals that run for a long ways are gut shot. When you rupture a bowel and the internal contents leak out into the abdominal cavity. The bacteria released cause a sort of toxic or septic shock to set in called Peritonitis. The muscles will become ridgid and hard within a few minutes and that animal will not feel like running or walking anymore. Left alone it will bed down and lacking the medical attention available to us humans it will most likely be dead in 6 to 18 hours. Even though he doesn't feel like it the animal can and will still get up and run during this time if you spook him. This would be an idiotic thing to do however. He won't die any faster just farther away from where you originally shot him.
I don't care who Jim Zumbo is or what he thinks he knows. If he says imediately chasing a gut shot deer is wise move then he is full of $*#%. And you can take that to the bank.