The elk was able to soak up quite a bit of punishment - 4 shots to bring it down. You were able to see the bullet tracks in the mist/air and the first went into the belly, the second and third were similarly poorly placed, and only the fourth went into the heart/lung area, dropping him. Luckily for the hunter the elk stood there and took it. It it had bolted after the first shot things wouldn't have turned out so well.
I think you said all that needs to be said right there. No matter what you shoot if you put the bullet in the belly you aint gonna get the animal no matter if it's a coyote or an elk.