Since you list the .375 I take it such a medium fits your description of a "big" bore.

First, there are lots of options for reduced loads in the .375. Using 5744 powder, the .375 becomes a 9.3x57 and that is a compliment, not a criticism.

Second, you haven't mentioned what you want it for, only mentioned North America.

Well, as far as North America is concerned, based on your criteria, I'd buy a Ruger M77MKII in stainless with synthetic stock in .30-06 caliber.

Then I'd have a gunsmith toss the barrel and screw on a 9.3x62 barrel, add good sights and a Leupold 2.5-8 scope.

After you did that you'd have the best North American medium you could buy in my opinion.

You'd have the many 9.3 bullets, and if you bought some bullet sizing dies, you could use any .375 lead-core bullet, too, sized to 9.3.

With a 24 inch barrel, run full throttle, you'd have 286's at 2400 fps at least, and downloads to .35 Remington-type energy performance would be a cinch.

To be honest, for a NORTH AMERICAN rifle, I really see no need for anything over .375 caliber unless you restrict yourself to shooting big bear. For an all-round, I like the 9.3x62 idea.