A .375, a .416, or a .458 will work with good bullet placement, just like any other cartridge. For that reason I have come to think that the question of a suitable cartridge in the context of North American game and for that matter in the case of African planes game is pretty much irrelevant; provided we can agree on a lower limit which I define as a 6.5. The proper cartridge for any given game is the one which provides a reasonable expectation of killing the animal, with a single shot, from any angle, within the range limitations of the cartridge and the hunter. With the correct choice of bullet, all modern cartridges of 6.5 caliber and larger, that are commonly chambered in factory big game rifles, are perfectly suitable.
You must first determine the type of country you will be hunting in and decide on a choice of rifle and scope combination that is best suited for hunting there. The physical size and weight of the rifle, needs to be considered, and this will impact the choice of cartridges you have to choose from. Your choice of sight should be in context with the likely shooting range, the size of the target, and the amount of light that is typical of such hunting. Most hunters tend to over-scope their hunting rifles; choosing glass that would perform well at 1000 yards for shooting at 100; which due to poor light transmission and too small a field of view all too often will cost them the shot. Besides, big scopes on small rifles look ridiculous.
Every rifle manufactured today is available in a selection of very similar chamberings; simply choose the one that flicks your switch. It that choice is a 6.5X55 or a 7X57 Mauser in a light Mannlicher stocked carbine, all is good. If your choice is a .378 Weatherby in a Mk-V, that too is good. Is one better than the other? That depends more on who is doing the shooting, than it does on what is being shot.
The most important question once you have acquired your rifle and scope is to decide which bullet is the most compatible with consideration given to the game being hunted, and the range it is likely to be shot from. I have come to appreciate the qualities of heavy for caliber round nose bullets, particularly 220 and 240 gr in the .30/06, 350 and 380 gr slugs in the larger .375s, and 550s for a big case .458s; but that might not work for you if your shots are on the long side of 250 yards. I've shot .375/380s successfully out to 400 yards, but 2' of hold over is a challenge and since I don't hunt with a range finder the chances of me risking such a long shot on a live target, with that particular combination, is highly unlikely. In recent years, the Barnes TSX seems to provide the answer, regardless of what question is being asked. While I prefer lead core bonded solid shank bullets, I always seem to have some TSX/TTSXs seated on a few rounds.