I have at least one Mk V in all of the calibers mentioned except for .460 and .30-30. The .378 recoil with a brake is not, at all, murderous. It is very similar to the .340 without a brake. The only issue would be ammo availability in a pinch which is why, on many blogs, you will hear the praise for the .375 Weatherby because it outperforms the .375 H&H and, if you run out of ammo it can shoot the .375 H&H. When you shoot the .375 H&H in a .375 Weatherby the casing is fire formed into .375 Weatherby.
Chris is on point but, on the push feed, I have never had a problem with it. In a few months shy of 70, the push has always served me well. And, I have nevr had a jam in a Weatherby Mk V. So, to me, "control feed" is just a marketing and promotional vehicle/ploy so that the control feed rifles will sell.
What happens when a control feed jams as opposed to a push feed? If it grabs the cartridge so tightly what will the answer be.
Recently, on a TV show I think was called Hunting, a brown bear hunt in Alaska was covered. The hunter was an idiot who, at first, wanted to hunt with his bow even though he couldn't hit the broad side of a barn with it. So, the guide required that he use a rifle.
When this guy shot the brown bear he also screwed up that shot and that brown bear came at them like gangbusters and with blood in its eyes. The first guide's brand new rifle, with a Mauser 98 action, jammed and, but for the other guide, they all would have been like that guy they called the Grizzly Man, dead as a door nail. So much for the Mauser action.
So, the .378 Weatherby is a great round and a great gun so long as you can shoot it properly. Just be certain that you send your ammo to your PH with enough time to confirm that it was received in good condition. Then take another box or two with you. When I go to Alaska, I pack 2 boxes of ammo in my rifle case with my Mk V. Just be certain that your case will accommodate what you need to pack in it.