The 416 Rem. Mag. is more efficient than the 375 RUM. It can drive a 350gr bullet at 2700 fps with less powder than the 375 RUM uses to get the same speed with a 300gr bullet. And with 2" less barrel length. Plus it has about 25% more cross-sectional area = more shock.
I was getting 2850 fps when I loaded 350s in my 24" .416 Rigby. As for the .416 Remington being more efficient than the .375 Ultra, you're not comparing apples to apples; a .416 bullet that has the same SD as the 350 gr .375 would weigh 430 grs. How fast could a .416 Remington push a 430 gr bullet, even if you could find one? Probably about 2400 fps, similar to the velocity I get with a 380 Rhino in a 22" .375 Ultra, and about 100 fps behind what I get from the .375/350 gr Woodleigh.