One of the original selling points of the .375 H&H when it came out in 1912 was that a hunter could buy one and, by using bullets of 235, 270, and 300 grains, hunt anything anywhere in the world. This was fine in theory, and no doubt many did it, but I’m equally certain that a sizeable percentage of nimrods ended up as lion dung or re-shaped into something the size of an elephant turd because they used the wrong bullet on the wrong beast.
If you want to make the .375 H&H a multi-purpose cartridge, you stick with one bullet weight—the 300-grain—and you limit yourself to large and/or unpleasant animals here and in Africa. The .375 H&H is generally regarded as the .30/06 of Africa, the single most useful round for all of that continent’s vast menagerie of game. But not, please, courtesy of a whole variety of bullets and bullet weights.
Some rounds are advertised as being good for varmints and big game. In my youth, both the .243 and the .244 (later the 6mm Remington) were touted as such when they came out. I did a lot of varmint shooting with both, and found that both were mediocre at it. Compared to the .22 centerfires, they had too much recoil and not enough accuracy. With the bullets available then they were OK on deer, but no better. I haven’t owned one of either in a long, long time.
No cartridge has been used for more purposes than the .30/06, and it remains our best multi-tasker. With 150-grain bullets that can be driven at 3,000 fps, or very close to it, it is sudden death for light animals like antelope and small ratty deer. With 180s and 165-grain all-copper bullets, it can take almost any big game, and if you handload it with 200-grain slugs that say Swift A-Frame or Nosler Partition on the box, it can drop very big game as well. If you’re looking for the single most useful bullet weight, it’s probably still the 180.
But the .30/06 is not a varmint cartridge, despite some desperate attempts to make it into one. This multi-tasking stuff goes only so far.