I read a fairly extensive article on this cartridge in the this rifle (the Weatherby you are talking about). Unfortunately, I don't currently remember which one it was.
I think that the .338-06 cartridge should be great for what you are talking about. I keep on thinking that one of the major gun manufacturers (Remington or Winchester) would come out with a .338-06 of their own, but so far only Weatherby has. Frankly, I don't know why the .338-06 isn't as popular as it could be.
However, I am sure that you realize that your ammunition selection will be extremely limited, unless you handload. If you don't handload, then you would be much better off with a .300 Win. Mag or a .338 Win. Mag. which have countless types and brands of ammo.
I was going to get this Weatherby rifle in the .338-06, but the article posted sub-par accuracy with the 210 Noslers (the only type available). I think that the factory ammo posted something like 2" groups at 100 yards, with some handloads a little better, while some handloads were a lot worse.
I know that you don't need a gun to shoot better than 2" groups at 100 yards if you are hunting something as big as elk, bear, or moose, but come on...for over $1,000 for this rifle, I expect better accuracy. This is the other reason why I didn't buy it.
I wouldn't buy the gun yet. I would wait until Remington or Winchester offer their rifles in this cartridge, and hopefully manufacture their own factory ammo. Until that time, I would look into something else.
By the way, I have a .338 Win Mag with a muzzle brake that kicks LESS than a .270. This gun is a Winchester M70 Classic Stainless and will group PMC 225 grain Barnes-X bullets into about 1" at 100 yards. Granted this rifle is a little bit heavier than the Weatherby ultralight, but when you compare all of the pros and cons, this rifle was the best over the Weatherby Ultralight in .338-06 hands down. (Yes, I know that muzzle brakes are loud, but you can wear little hearing plugs and it won't make a difference.)
Hope this helps.
Zachary