I've killed a few Bulls with my bow, but not rifle yet, so this post is not totally applicable to the OP question. Call this boredom and appreciation of this thread as I am on another god forsaken BAR study break. Someday this war is gonna end ...
So, here goes. I started rifle hunting pigs in CA with an 06. All the old timers told me that was the only one I needed, so I only had an 06 for about 15 years. Killed Deer, Pigs and exotics with it and only shot core locts. When I did my part, it was over quick. But, elk hunting was always with the bow OTC as I cant draw a damn rifle tag to save my life ... and I was an idiot bow purist for some time. Anyhow, the 06 never got to hunt Elk, unfortunately (later converted to a 35 wheel). Then I wanted something lighter to hunt high country mule deer with, so I bought a Rem 700 Mountain Rifle in .280 Rem. Got lucky as it shoots far more accurate than I do. Kills everything I hit. For whatever reason, that rifle is an extension of my arm. I shoot it offhand in hunting scenarios really well, which is nothing short of a miracle given my propensity for buck fever or bouts of instant insanity when I see game. But, It Missed a few when I had too much time to set up and think about it. So, I thought I needed a bigger round to knock down the mountain boars where we were hunting on longer cross canyon shots (it was the cartridge's fault, not mine, you see).
So, Then I bought a Rem .300 RSAUM from an old hunting guide I shared some time with. He said it didn't shoot well, so I got a deal on it. It had a brake on it and it was the loudest gun on earth, but, I killed a pile of pigs with it, including some real bruisers, and it knocked the hell out of them. My longest shot to date on an animal was with that gun (468 yards on a 250 LB nasty old Boar, which is as far as I will shoot at game). Turns out that rifle just needed cleaning ... and (2) sets of ear protection.
Then I was convinced by the internet that I needed a big round to buck the wind and knock down Pigs (and hopefully Elk) at distance. So, I bought a .338 LaPua and sold the RSAUM. No brake. Not terrible recoil, and it shot accurate as heck. But, it never left the truck as I didn't like carrying it (a big factor for those considering magnums with 26" barrels that you only learn once you are in the field). Then I bought a .300 Win Mag, which also never left the truck. Same issue. Sold both.
Long point being, the .280 Rem is always with me, and it has killed a ton of animals because I am so comfortable with it. A 140 grain TTSX or 139 GMX traveling around 2900-3000 FPS out of the barrel with good shot placement is all I have ever needed. When I do my job, everything dies. Based on the amount of big boars and Deer it has decimated, I have no desire to reinvent the wheel when I finally do draw an Elk rifle tag. So, I agree with several of the posts on this thread. A giant caliber won't stack more game. A reasonable rifle/cartridge platform you shoot well absolutely will. A 6.5 CM in the right hands has and will kill Elk, or other big game as well as a big magnum. But the key to all these discussions is the concept of "reasonableness." If you shoot a CM really, really well, and you shoot a good bullet, you are fine. If you shoot an 06, or a .30 cal magnum really well, that'll do, and likely a bit better based on math. But it is by no means a necessity. I've taken several folks hunting that had new whiz-bang $5,000 super mega magnums that resulted in scope cuts and wounded animals. A reasonable rifle/cartridge would likely have solved that.
All the East Coasters or new western shooters I see posting on this site (not being disparaging, at all, as I like the interaction) should really be considering a rifle/cartridge that Is reasonable for the game you are chasing that YOU shoot really well. If that is a 6.5 something, a .270 something, or a 7MM something, so be it. Just shoot straight, and use enough bullet. I have no problem making fun of myself for thinking I needed more. Just please don't make the same mistakes I did.