Realistically the .338 is overpowered for deer-sized game.
It tends to thump worse than the .375, too, in my experience, probably largely because the .375s are usually quite a bit heavier.
That said, the .338 is a wonderful round for medium to large game, excellent power and good trajectory out to realistic ranges at which you might hunt some of the larger game.
My biggest gripe against all of these magnum calibers is the belt. The belt used to be there for headspacing purposes. It's not needed. It probably wasn't needed back then, either. Some of the new calibers are dispensing with the belt and losing nothing in the bargain.
If the largest game that I would hunt would be deer, I'd not choose the .338. If I were hunting mule deer, elk, or moose regularly, I'd very likely get a .338 and be very happy with it.