I'm building a 376 Steyr on a Vz24 Mauser action with a 22" douglas barrel, but I haven't worked up loads or shot the thing yet, so this is all based on what I've heard and my own figuring using Load from a Disk and info off the web.
I like the idea that this round was designed from the get-go as a dangerous game cartridge for a shorter-barreled rifle. From the looks of it, it appears to me it was designed to load and extract reliably.
The 376 Steyr should be an extremely useful and flexible cartridge. You can load it up tp near 375 H&H ballistics: Hodgden's website shows 2,410 fps with 62 grains of Varget and a 300-grain bullet, and 2,651 fps with the 260-grain Nosler partition. Some sources say that 2,500+fps is achievable with the 300-grain bullet and careful handloads. This range of velocities with a 300-grain premium bullet seems to be the standard big and dangerous game load for this caliber.
In a 10-pound gun, shooting a 300-grain bullet at 2,410 fps recoil is 32 foot-pounds, quite a bit less than the 300 H&H becuase you're only burning 70% as much powder. In an 8-pound gun, recoil is 40 foot-pounds, about the max I'm willing to tolerate. With a 6-inch kill zone, the rifle will sight in to 200 yards and have a point blank range of about 220 yards.
I expect to work up a handload with a 375 Win bullet like the Hornady 220-grain flat point to something around 2,400fps ( a little hotter than a 375 Win). With a 6-inch kill zone, you can sight in at 190 yards and,again have a point blank range of about 220 yards. Recoil in an 8-pound gun is only 20 foot-pounds, about the same as a 30-06.
Since I don't intend to hunt elephant or rhino, I'm looking at this as the big gun in my hunting battery. In a pinch, it could serve as the only gun on an african or Alaskan safari, with a range of handloads to cover everything from plains game at medium distances to dangerous game, in a rifle that is small enough and light enough to poke around through the brush with.
As I warned in the subject line, I could easily be wrong and will respectfully defer to anyone who has actual hands-on experience. I hope this gives you some ideas to chew on. YMMV. Good luck!