These are the forgotten cartridges. They have maximum loading that rival most of the newer shot shot cartridges and the re-loader can tailor his load to meet the needs.
I have and shoot a .375 H&H, Ruger Number 1. For years, until last year when I couldn't hunt at all, used my Ruger for deer hunting. I don't call it over kill, not with 285 grain cast lead at 1865 FPS. My philosophy of One Shot One Kill is enforced with having and using a 'falling block' single shot rifle.
I think that your 300 H&H would be a better choice, a much better selection of much cheaper bullets. You can go with the extremes, 110 grainer at 34 to 36 hundred FPS and 250 grainers at an easy 2500 FPS or load them back to 30-30 specks with the same bullet.
Back in the early 70's, .375 brass was a special order ordeal. Now we have on-line suppliers for both brass and bullets.
Enjoy and be safe.