Thirty years ago I bought a .270 to serve that purpose. It'll work on elk and lots of folks still use it on moose. Big bears I always left out of the calculation, figuring I'd buy something specific for that, if and when. The benefit of that approach is that I had a great cartridge for the 99% of the hunting I normally do, without having to have my head kicked out from under my hat by recoil.
With your choices, I'd still go for the 99% route, and a .308 would fit it well. It's likely more accurate and certainly lower recoil than the other choices, and both would help encourage practice. Very low-cost mil surp rounds are available, which again encourages practice. Also, I don't think an elk would be able to tell the difference between a .308 and a 30-06. The .338 is a fine round, but is much more than I, at least, need for my hunting.
There's been a big move afoot in the last couple of years to declare the .270/.308/30-06 marginal for elk. I've stared at the keyboard for awhile now, trying to think of a polite synonym for "hogwash," and am having a little trouble, so just let it go. If you want a .338, by all means get one. If I like a rifle, I practice with it. If I just like a rifle and am not convincing myself that I need it, I can get rid of it when I no longer like it. Sorry, philosophy mode off.
As to rifles, I concur with your choice. I recently generated a liking for an unneeded 300 Win Mag in a new Winchester/USRAC Model 70 Classic LT, wood and blue. The store didn't have a stainless there in wood, or I might have bought it instead. This is a really fine rifle. The stock is nicely done, the checkering wraps around the slim fore-end. It weighs 8 pounds with its 26-inch barrel and tames the recoil well. The blueing is nice and the trigger adjustable. I won't finish wringing it out until after hunting season, but I'm now a believer in the M70 - it's a quality rifle. Initial indications are that it might well be an honest MOA rifle, out of the box. They don't sell the Classic LT in short action, though, so a .308 would move you over to the M70 Feathrweight.
Good luck,
Jaywalker