Their are so many things to consider when killing animals...angels,distance(how good of a shot you are)the rifle you use, the bullet choice.....I would not preach to anyone to shoot at the shoulder of an elk....unless you know were to aim(can shoot) and have a weapon capable of penatration and also a bullet that will stay together....also a 12" circle is pretty good shooting, but thats a big group size to be aiming at the shoulder at that distance(consider the lungs).....now if you keep practicing and get to were you can shoot 3" groups at 300 yards, then you have a chance if you have a gun capable of penatrating/and bullet....Thats still going to be risky since elk are so thick and big boned...I would forget about the shoulder after 200 yards and aim at the lungs...
I shoot crop damage permits for whitetail, and have killed more deer than i can remember...30 to 50 a year..Some have to be shot so they will run in the woods , others you have to put them down in their tracks....Farmers don't want the carcass in their fields so you can save yourself alot of dragging if you shoot them right......I usually aim at the center of the shoulder and a little low (beware of high shoulder shots)on deer i want to drop inn their tracks, or the head/neck/spine....I have missed the heart completly on alot of shoulder shots and still dropped them in their tracks...I beleave the shock of those shots is what puts them down...The heart is usually like gello...and if you look at the location of were the bullet entered and exit, alot of times you realize that you did't even hit the heart, So it has to be either the shock or you hit the main vessol going from the heart to the brain(the aorta i think its called)...(also I try to hit both shoulders)perfectly broad side deer with perfectly placed bullets will fall 95% of the time...I have had quite a few run on ones that i missed hitting both shoulders....Grant it killed them, but they run....
On ones we want to make it to the woods we shoot a non expanding bullet thats mid weight and aim at the lungs...they can run several hundred yard shot like this...If you shoot them with a bullet that expands greatly some will fall just like you shot them in the shoulder/head/spine/or neck...also liver shots work well to let them run, but I don't usally aim for it because it takes a while to kill a deer, and i don't want to see something suffer....on quartering aimals you have to aim at your exit hole and know your going to penatrate the vitals..A elk quartering at you would be tough...I think i would wait until i had a better shot or shoot the neck...I want an exit hole on shots you think they will run on .....I hope this helps but, alot more expeirenced elk hunters could probally teach you better than I....I would think a good elk guide could teach you more than you could learn inn a lifetime on your own....