I've shot a lot of deer sized game (including several WTs) with a 375 H&H. It actually does a lot less meat damage than the smaller bore fast magnums. The 375 H&H has always been a round known for penetration so it fires fairly heavy bullets at modest velocities to get deep into very large game. So when you shoot smaller game with a 375 H&H you tend to get a .37 caliber entry hole and about a .39-.40 caliber exit hole with no bruising or bloodshot. That big slug just plows though and keeps going. In Africa they have a saying when it comes to game shot with a 375 H&H: "you can eat clear up to the bullet hole."
I use a 300 gr bullet for everything in my 375 H&H and it has always performed well even on game like impala and warthog, both of which are close to the average WT in size. They sell lighter bullets you can push faster but that negates to whole reason the 375 H&H exists. If you want to shoot a lighter bullet that moves faster and causes bruising and bloodshot get a 338 Win Mag or a 340 Wby Mag.
There is a reason the 375 H&H has always been called one of the few "perfect" rounds. You can literally take everything on the planet with it and you will rarely be over or under gunned. For the really big and little stuff all you need to do is shoot solids and use soft points for everything else. My rifle shoots handloads softs and solids to the same point of impact and I always carry a few solids. I even shot a jackal with a 375 H&H once with a solid and he had neat holes on both sides and the skin tanned out real nice.