Interesting you would say that. I was drawn to the double rifle and 375 H&H due to the nostalgia associated with both. The more I read threads on this forum, the two together may not be the best combination depending on the quarry you are hunting.
As you stated, the bolt action seems to be a better fit for the 375 H&H. Taking advantage of its longer range capabilities, I can see the 375 H&H as a very effective North American big game rifle and wouldn't hesitate punching an elk inside of 450 yards aided with a quality scope. I could extend that logic to plains game as well.
My double rifle caliber has been evolving to the 9.3x74r but the 450-400 NE is intriguing as well.