He should have backed out because the deer would have layed down and bleed out on the spot. By being pushed, you run the risk of limited blood trail or driving it into an area that is either non accessible, or to thick to find anything, etc.

I shot my buck Wednesday and thought I had a good hit. Got to my arrow and found blood but also stomach contents. I saw him walking away and watched him get to the top of the hill. I left and came back 3 hours later. there he was dead as a doornail laying almost where I last saw him.

Had I pushed him, he may have gone into posted property that I have absolutely no access to and I would have lost the deer.