Although Remington's quality has gone down in my opinion, I personally wouldn't say that their rifles are necessarily bad. As far as the .338, .375, and .416 are concerned, I'll address each separately.
1 .338. This is essentially an elk cartridge. Sure, you can use it for bear, and even mule deer on a combo hunt. Recoil is much more than a .300 Win Mag and it can take an elk almost as effectively as a .338 with the same bullet placement (through the lungs). I own a .338, but I use it for really big hogs and perhaps may use it for elk some day. I own a Winchester Classic Stainless in this cartridge.
2. .375H&H. This is the minimum caliber for dangerous game in most African states and thus is probably the most popular. I'm thinking that many people have fallen victim to the hype about how you need a controlled round feed for dangerous game (and I admit that I was one of those people). So maybe people who buy .375s usually buy Winchester. I own a Winchester Classic Stainless in this cartrdge also, although it has been heavily customized. It is, surprisingly, my most accurate rifle.
3. .416 Rem. This is a great cartridge with no where to go. When it was first introduced some years ago, the thought was to fill in the gap between the .375 and the .458. It sounded like a good idea but, for whatever reason, it wasn't a big seller. Perhaps the controlled round feed thing also played a factor. I do not currently own a .416 Rem Mag, and I don't think that I will, especially now after the news.
Zachary