Having hunted the big and dangerous guys in Africa a few decades ago,, I used a 375.........once! IMO, anything big, with claws, teeth, or horns at spitting distance with a 375 may be asking for trouble, IMO. WildRose, likewise, buddies in Africa also use the lighter calibers for the plains game.
With thin skinned game I agree with Yotekillers statement. My LR hunting is defined by maximum ranges for antelope to 1200 yards, deer to 1000 yards, and elk to 500 yards. While I have used the big 30's, my caliber of choice for the last 10 years or so is a 6.5 run at 3000FPS using 140/142 Berger/JLK VLD's. Driven at this MV, and shots surgically placed in the vital zone, will penetrate(sectional density) and shed energy that has proven to be more then sufficient to drop dozens of game animals from 100 yards to the max ranges stated above. Running higher velocities can expand the max distance performance, but trades off the more likely shorter range behavior of the bullet. I did a lot of testing(observing actual kills) to determine the sweet spot. I'm very picky about the shot, and conditions. Using calibers from .264-.308, it's all about the bullet, and it's behavior at point of impact. While I have been able to achieve comparable terminal performance with larger calibers, the diffetence in actual killing performance is not enough(using my range limits) to trade off rifle weight and recoil control which influences accuracy, and the ability to spot hits.
It seems, reading through the 200+ posts in this thread, that each individual has their own experiences, circumstances and criteria. It's wise to be cautious of extreme positions and gross generalities. IMO.