I lived (and hunted) in Tanzania from 1970-1973. I started hunting with a 7 mm Rem Mag when I first arrived, but Jon Speed told me one day he had seen a Win Mod 70 African in .375 H&H in a gun shop in Tanga. We flew up and took a look at it, and after waiting for a license to come through I picked it up for around $200, IIRC. I have never regretted that purchase.
I used it on everything from dik-dik to cape buffalo, with a lot of plains game thrown in. And when I ran low on ammo on safari in the north of Tanzania, I was able to pick up several boxes at the local gun shop in Arusha, which would not have carried any of the wildcats that people here are telling you are better cartridges. Perhaps on paper they are, but in rural Africa, they are not.
My longest shot with that rifle was around 500 yards on an antelope of some sort -- probably a Grants gazelle, as we shot a lot of them for camp meat and leopard bait. Most shots were in the 70-150 yard range, and I only remember once having to use a second shot. That was on a zebra, which are notoriously hard to kill and will run for miles if not dropped with the first shot.
I also carried a .450 Nitro Express for a while, but only for cape buffalo or elephant. And indeed, the .375 H&H was my preferred rifle for buffs. I once had to dispatch a buff in thick grass after someone I was hunting with managed to wound him. The .375 put him down right now!
Hunting near Tarangire the last year I was in Tanzania I broke the stock behind the action when I tripped on a root chasing a herd of buff. I epoxied it next to the camp fire that night, wrapped it with the only electrical tape I had (white), and was hunting with it the next day. I have a picture of myself holding that taped-up rifle, with a nice oryx that I took the next day at around 300 yards.
So yes, there are more efficient cartridge designs, and there are probably better rifles than the Win 70 African with a wood stock, but they will not kill game any deader or so far away you cannot see if they're a trophy.
And when I returned to Ohio and hung a metal target at the far end of my farm, my .375 H&H was my best choice to ring that gong nearly half a mile away.
My vote would go to the .375 H&H. Fred