It's common for people to use the .375 H&H for African plains game including the smaller antelope, and that cartridge doesn't destroy them... but they're using a typical 275 to 300 grain bullet, not a 250 grain bullet. No reason a .338 with elk-capable bullets should do more damage. I'd be a little wary about using bullets under 200 grains for close in shooting unless you are sure you'll be making those broadside lung shots and not have to deal with a shoulder. A 180 grain bullet will be sizzling out of the .338, and within 100 yards will be more destructive if you hit the shoulder blade... all that energy is going to do something.
I like a well-designed 225 grains for the .338 as more than enough for deer and elk at any reasonable (don't get offended, think 'traditional, which means 400-450 yards or less). Shoots pretty flat, hits very hard, not too destructive but certainly destructive enough. The Goldilocks bullet. I have a case of Winchester 230 gr FailSafe ammo from my dealer days, and it will do if I will do. Hornady loads a 230 gr ELD-X at around 2800 ft/sec and that's plenty for deer and elk out to at least 700 yards. I like the sentiment of using one load (cartridge, bullet) out of a big game rifle instead of trying to optimize unnecessarily, and IMO this is the sweet spot bullet weight-wise for the cartridge.
I personally find the .338 Win Mag is a surprisingly comfortable cartridge to shoot, more so than a .300 Win Mag.
