I'm partial to any of the 30's. But most of the 7mm's or 270's will do the job. It is really more about bullet choice than caliber with anything from 25 caliber up to the 30 calibers.
I can understand the argument for the 375, at least on paper, but let's be realistic. They are simply overkill for everything 99.99% of hunters are going to actually hunt. The only game in NA where they would be of any advantage is coastal brown bear. I've priced some of those hunts, they start at about $20,000 and go up to $40,000. And that doesn't include tips, transportation, and taxidermy bills. Anyone who can afford the hunt brown bear can afford more than 1 rifle. Even for 90% of African hunting they are not needed. There are only 4-5 species of game on the planet justify rounds larger than 30 caliber, and only a handful of folks will ever hunt them. It's like owning a 1 ton dually truck rated to tow 30,000 lbs when you're never going to tow more than 3,000 lbs.
Like virtually all hunters elk is the largest animal I'll ever hunt. My 308 weighs under 6 lbs scoped. It is a joy to carry, is cheap to shoot, extremely accurate, and with virtually no recoil. I can sling 130 gr bullets over 3100 fps, 165's @ 2800-2900 fps and 200 gr Accubonds @ 2500 fps. That's a 300-400 yard elk rifle, I don't care what anyone says.
The 30-06 gives you 100-150 fps more speed and the 300 mags another 200-300 fps for a little more effective range. But that speed doesn't kill them any better at closer ranges.
I have no doubts the 338 and 375 mags can offer a slight advantage on elk or moose size game to make up for poor shooting. But from 9-10 lb rifles vs 6-7 lb rifles. And with 40-50 ft lbs recoil vs 15-20 ft lbs recoil in a 308 or 30-06, 25-30 ft lbs with the 300 mags. They don't kill 'em any deader if the bullet is in the right spot.
As said earlier, I'm partial to the 30's, but most of the 7mm's including the 7X57, 280, 7-08, any of the 7 mags and even the 270 are very close. Just not as many options for bullets heavier than 175 gr. Probably not needed either.