I have never bothered myself with the thought of a dangerious game cartridge. I guess the question is still, what do you call dangerious game? There is no animal in North America I would back down from with either a 338-06 or 338 win mag and a good 250 gr bullet. Here's my thinking. If I were to hunt such an animal, a guide would be required. He has the stopping gun. what I want is a rifle I shoot well that will put a bullet capable of making the trip in the right spot with the first shot. That will be your best shot, make it!
i have never shot anything bigger than a 350 rem mag and that thing was brutal, mod 600 Rem. I would not care to bet my first shot on a rifle I didn't like shooting. I've never shot anything bigger and next down would be the 338 win mag. I cannot imagine any animal taking a well placed first shot with a good bullet and going very far. 250gr bullets were really bad for recoil but manageable. I think I'd try to old Speer 275 gr if I could find them and shoot a lot. You have to learn to handle the recoil. lacking that bullet, I'd fall back on the 250 gr Hornady SP or the 250 gr Nosler. Either bullet at around 2700 fps would be very formable.
You could go to bigger bullets and larger cartridges but your cartridge is only as good as you can shoot it.