I don't think Elmer Keith would use a jacketed .44 bullet, if they even existed back then.
In both books he specifically mentions that the "part jacket" from a Remington .44 bullet was found in the deer by his son, Ted, confirming that one of the hits was from his gun since the other shooter was using a .300 Magnum rifle.
Lets say an experienced hunter and shooter in the rifle section with a open sighted 444Marlin asked everyone about it's effectiveness as a long range mule deer rifle, up to 500yds. He would politely or not so much be scolded for his inquiry.
He would, indeed. However, Keith wasn't arguing that his pistol was effective as a long range deer hunting firearm, only recounting an incident where he helped put down a wounded animal at long range.
Also, it needs to hit the vital area, not just anywhere on the whole wounded 200lb adrenaline soaked Mule deer.
And that's a critical difference between claiming that a particular gun is an effective choice versus claiming that it could happen. Keith never claimed that he aimed for the vitals and scored a hit where he aimed. The story clearly indicates that he was aiming for the deer, not some specific point on the deer, and wasn't even that sure he hit the deer until the jacket fragment was found in the carcass.
By the way, for what it's worth, the story indicates that at some point before Keith fired his last shot, Kriley ran out of ammo.
I didn't say Keith lied, or that the feat is impossible, just that there is no way you could do it without "luck", or random chance.
Keith mentions in his autobiography that he had done some shooting (witnessed by Judge Don Martin) at about 500 yards (distance paced out) "just before hunting season" with the particular revolver in question and was able to put 5 out of 6 shots on a rock that was "about three feet long by about 18 inches high in the middle, tapered a little bit at each end".
I do believe that there was a significant component of luck in the shots at the deer, but it's important to understand that Keith was no stranger to long distance shooting with a revolver and had used this specific revolver at long ranges not long before the shots were made.
What about wind drift, how much would that factor in? Could be monumental. I think much more than a rainbow like trajectory. How much bullet drop would we be talking about?
Remember that Keith claimed that Kriley spotted at least one shot for him, telling him that he saw the bullet strike in the snow through his scope. That would go a long way toward explaining how Keith compensated for wind and drop.
It's also important to remember/understand that Keith used a method for long range shooting using open sights that virtually negates the issue of drop if used properly. He explains it in his book. As long as the gun and shooter are capable of the accuracy required, the process can theoretically be used to score a second round hit at virtually any range if the shooter is adept at the process and the location of the first hit is known/spotted.
I have not had the opportunity to shoot a handgun past 300 yards, but the technique worked well out to that distance, subject to my limitations and those of the pistol in question.
There is no way a 44 magnum would penetrate the body of a full grown Mule deer buck at 600 yds. 600-700 fps is not going to do it.
It can be a real eye-opener to realize how much a heavy, slow-moving bullet will penetrate, especially if there's little to no expansion involved.
Also, you may be overestimating the size of the deer. According to Craig Boddington in his book "The Perfect Shot North America", the average mule deer buck taken in the U.S. is about 150lbs. A broadside shot wouldn't need to penetrate very far to shoot through a 150lb deer and the 16" to 18" of penetration I would expect from a 600-700fps .44 bullet, moving too slowly to expand, would likely do the trick if a heavy bone didn't get in the way.
Keith didn't make any quantitative claims about the buck's size or weight in either source I have for the story.