Good bullets can certainly make a difference, but Im not convinced they are required until the velocity goes north of 2800 or so. The reason I believe the bigger is better is the wrong advice is that the truth is most guys just cant shoot them. Im one of them. The fact is all of us can shoot lighter recoiling rifles better. I've been guiding full time for 33 years now and I see it every single year. Ask any guide with any experience and he will tell you he loves to see a guy get off the plane with a well worn 270 or 30/06. Chances are, that guy can shoot those guns.
When it comes to Yukon bison, I have personally killed four. Two with a 338 Federal, one with a 358 Winchester, and the last one with a 338 Ruger RCM. Only one needed a 2nd shot, and that was the last one. The first shot would have killed him but he was bedded on the top of a steep ridge and I didnt want him to move. He didnt. I guided for bison in the Pink Mountain country in northern BC back in the 80s and saw many guys take them cleanly with cup and core bullets.
A few years ago Fish and Game charged a first nation guy with hunting bison without a license or tag.They also charged him with using an inadequate caliber. He was using an old model 99 300 savage. They got him on the hunting without license and tags, but the inadequate caliber charge didnt stick since he had killed four bison with that old gun.
When I was a kid the First Nation hunters routinely killed moose with a 22 magnum.
