Since the thread title mentions "big game hunting" this is where all the old timers who claim to have killed hundreds of elk with a .243 Win and no tracking chime in that it's all about shot placement and only one shot is needed. But before I digress too much, most every repeating rifle I can think of will give you a 3+1 option but there's no guarantee that 4 rounds will be enough even with good shot placement. A couple of years ago a coworker and his brother shot a bull elk three times in the heart/lungs at 300 yards with a .300 Win Mag load and he still managed to go 50+ yards and almost fell into a coulee before he finally dropped. They had good bullet expansion, three good sized permanent wound cavities but the elk didn't know or care. The point of that story isn't that the elk needed three rounds but rather that for some, the difference between 10 yards and 50 yards might mean 1 hour to pack the meat out or 12 hours depending on where the animal drops. I'm sure that folks will be along shortly to tell you that elk aren't tough and that all you need is a 6.5 Creedmoor and one round with good shot placement for one-shot DRT kills on elk out to 600 yards. I'll stick with my Talkeetna and 4+1 rounds of .375 H&H plus six extra rounds in a belt pouch since I like a statistical advantage commonly referred to as stacking the odds in my favor.
I'll go there. Shot placement really is key, regardless of what you're shooting, and regardless of the stories you've heard. The story mentioned above about the guy's friend who shot the elk in the foot is a great case-in-point of this phenomenon.
I'm not entirely sure exactly when the super duper magnum fad started with regard to hunting North American game, but it seems to me that it really took off in the late 1990's. Now, I have no problem with someone bringing a gun on a hunt that is a bit more than someone needs (or even a LOT more), assuming they can shoot it accurately. But, I roll my eyes every time I hear someone tell me how a .308 Win or .260 Remington can't do the job on an elk. More often than not I find those statements are spewed by people who simply can't hit what they're aiming at, and want to blame the caliber for the loss of game... they then tell me stories about how they (or their uncle's, cousin's, friend's, girlfriend) even lost an elk with a shot from a .300 Ultra Magnum just last season (surely the fault of the caliber, naturally).
Here's why shot placement matters:
If you hit and substantially damage a vital part of an animal that the animal can't live without, the animal is going to die. How fast it dies depends on how immediately vital the part is to the animal, and how greatly it is damaged. We're talking of shots that damage critical blood vessels, central nervous system shots, and/or vital organ damage. So, the ultimate goal when taking any shot (with any rifle) in hunting is to put it into a spot that quickly incapacitates the animal by damaging one of the aforementioned areas. In various hunting groups there's some debate on the best shot to take, but the standard "boiler room" shot on an elk is often chosen because it's a target rich environment for essential organs and blood vessels, and it's also a target of substantial size.
Nevertheless, if you gut shoot an elk with a .458 Win Mag it isn't going to go down faster than the heart shot elk that got hit by a .243 Win. That's reality.
With that said, the entire goal in choosing a bigger caliber is usually to increase your odds of reaching and damaging the vital areas of the animal. There is some value in choosing a more substantial cartridge for this reason! Some cartridges simply fire more substantial bullets, or bullets with better penetration capability. But, to suggest that a 6.5 Creedmoor is inadequate simply because other hunters have lost game with larger cartridges is conjecture at best.
And, I do completely believe your story about the heart shot elk that ran 50 yards. Honestly, it probably didn't matter if you shot that elk with a .300 Win Mag or a .50 BMG... he was going to run until he ran out of steam, and he did exactly that. I've seen this same phenomenon happen with humans who were pushed by adrenaline after being shot. In the most notable example I've seen, a guy ran 130 yards before dropping after being shot right through the heart. He was destined to be dead the second that shot hit him, and no hospital on earth would have been able to save him. But, he had enough survival instinct to run until he fell, and he did exactly that. Again, it wouldn't have mattered if it was a .38 Special or a .50 BMG in that case. He was dead on his feet, and running on borrowed time. The only guaranteed DRT anchor shot is a solid CNS hit (that's why it's the shot used by police snipers). That's obviously not recommended for hunting for the obvious difficulties in scoring that shot, weighed against the odds of wounding an animal that might limp away and die slowly and painfully. As such, the heart/lung shot is a more reliable choice in most cases. But, even as you and I have both witnessed firsthand, it isn't always enough to drop the animal (2 or 4 legged) in their tracks.