The .375 H&H was one of Jack O'Conner's favorite rifles. He allowed that it was a good plains rifle (antelope) as well as a fine mountain rifle. This from The Rifle Book, now likely out of print, & perhaps never available in French.
But it is not a bench gun, even for long range. I believe there is a new .375 Chy-Tac, so there may be some solid-copper .375 "match" bullets. But as far as I know, no solid copper Lost River bullets never won a benchrest match. They might be competitive with special barrels & rifling; something Barney Lawton was working on before he died.
But very expensive bullets. As far as jacketed bullets go, where would one even get good jackets? And if you could find jackets, no one can hand-point a .375. I don't know of anyone hand-pointing even a .338. Hydraulic assist only. Now there are some pretty good bullets made from hydraulic assisted presses, but they've never had the consistency of the mechanical presses, except perhaps for Berger, who I believe incorporates reasonably sophisticated sensors in their presses.
I've shot long-range BR for about 13 years now, ever since it moved beyond the Original Pennsylvania 1,000 Yard club. I like to believe I've a pretty good grasp of the sport. Of course, 1,000 yard competition has a much longer history in highpower. But in either sport, I don't believe anyone every seriously campaigned a .375 H&H. For good reasons.
Enjoy your hunting rifle. For a first benchrest rifle, either long or short range, go with proven components and equipment. Experiment as you will after you've gotten a feel for the sport.
Good luck to you.