I'm guessing by "hunting" you don't mean Long Range Hunting? The 375 Ruger has greater case capacity than the H&H and the 375 RUM has greater case capacity than the Ruger. If I was picking a 375 "hunting" round for large game, say out to 400 yds or so, I would pick one of these. My first choice of action for any 375 build would be a custom. After that it would be a Mark V or 700. In the case of the 700, I would replace the bolt and recoil lug.
A 378 Wby would top the heap of common factory cartridges but very spendy compared to the the RUM or Ruger. If you went with the 378 I would recommend a custom action first followed by the Mark V.
My smith who is currently building a 300 RUM on a 700 action for me is a big Mark V and 378 series fan. He has done hundreds of them.
If I was going to build that class fo 375, I would do a 338 Lapua Imp version on a custom action. I would not do a Lapua class cartridge on a factory action and if I did, it would be a Mark V.
For a LR 375, I would look at 408 Cheytac or something in that class like a 375 AM.