The venerable 375 H&H is my rifle of choice. I started hunting with it for white tail when I planned an African hunt. I wanted to be very familiar with the rifle I might have to rely on to stop a lion. I have brought home about a dozen white tail with it with a variety of ammunition. A broadside double lung hit will do no more damage than an archery broadhead. The difference is that the clean 3/8 inch hole through both lungs will drop the deer in its tracks instead of letting it run out of sight and need tracking. If you hit bone, the 375 will break it. A poorly placed shot will damage as much meat as a high velocity cartridge's projectile but a well placed shot does no more noticeable damage than a well placed 30-30 shot.
That said the 375 cartridge has abilities that lesser cartridges do not. As you suspect the cartridge has all the power needed to reach the vitals from any angle. So if your buddy wounds a big buck and it is limping away into a thick miserable swamp that you aren't likely to ever get it out of you have the option of putting a bullet in the heart even if you have to go through a hip to get there.
So don't rule out the 375 H&H for north American game, even though it works well on African game too. As for the recoil - if you load your own you can use projectiles from 235 grains on up to 350 grains. There is a lot of room to find and make a load that works well for you without any fear of hurting yourself.