I dinked around with a .338/06 that a friend of mine was building about six years ago. He seemed to feel that it was going to be the best all round rifle with moderate recoil, Blah, Blah, ...etc.
We ran it head to head with a field proven .35 Whelen (.358-06) and a vintage 30/06. All weapons had 26" barrels with twist rates optimized for bullet weights. All actions were pre 64 model 70 Winchester types.
Although capable of 1 MOA, the .338 rifle was noisier than the Whelen, and had more felt recoil than the 30-06. Ammo was more difficult to procure than the .35 Whelen, as the Whelen was by then a factory cartridge.
The rifle did not shoot as flat as the 30-06, and had no great advantage over either the .35 Whelen or the 30-06, so far as I could tell.
He had a devil of a time getting his des set up to expand the 30-06 brass necks without crushing the shoulders, but after I polished the expander button, the problem went away.
For my money, I would have gone with the .35 Whelen if I were after an -06 bigger bore wildcat. Bigger holes, you know... which is why I am doing up a .358 Norma Mag.
My friend was happy with his .338-06 though, and has taken several elk with it since then.