Have you thought about using a P14 or 1917 Enfield action?
There are "bubba-ized" p14 and 1917 Enfields floating around for $200 or so. These have had the ears ground off, and probably drilled and tapped for a scope so you're not ruining a collectible or restorable gun. I picked up a Remington-built 1917 that had been sporterized (poorly) for $200 at a gun show. Replacement magazine boxes are available from Brownells to accommodate full-size magnum cartridges that the home gunsmith can install. Dayton Traister has triggers, cock-on-open conversions and speedlock kits. I don't think there's anything wrong with leaving the bottom metal as-is - you can hold another round. Send the action and a P14 bolt (ready for magnum rimsize) to E.R. Shaw and they will install and headspace the barrel very reasonably. There's a good selection of solid wood, laminated and synthetic stocks available from Richards Micro-fit, Bell&Carlson, MPI and Boyds to name a few. You can have a custom rifle at a reasonable price, and do much of the work yourself.
It's a very strong action on which many DG rifles have been built over the years. My personal opinion is that a sound 1917 action is better than the Rem700: integral recoil lug, safety that locks the striker, mauser-style extractor, etc. IIRC, the Remington Model 30 was based on the 1917 action and was only replaced by the 700 because the 700 was MUCH cheaper to manufacture.
While the 375/338 (same as 375 Chatfield/Taylor?) will duplicate 375 H&H ballistics, you probably won't find ammo at the nearest hardware store in Alaska or Siberia or Africa should the airline lose your ammunition or you just ran out. I'm told 375 H&H can be found just about anywhere in the world. Plus it's a delightful round to reload.
Just my 2 cents. YMMV
