For hunting "ALL" big game in North America the chambering must be such that the biggest game sought can be easily handled. Everything smaller will die just as well with a heavier round but trying to use a minimal chambering on big beasts is asking for trouble.
For ALL animals including big bears I would likely go with the .300 Magnum and certainly wouldn't go smaller than a .30-06 or 7mm Magnum. If I planned to hunt the bigger animals regularly I might even go up to the .338 Magnum.
For hunting the lower 48 states there are any number of chamberings that would serve well but I would still choose something like the .30-06 because of the sheer size of moose, elk and buffalo.
As far a rifle, I am partial to the '98 Mauser action. For many years my only big game rifle was a .30-06 built on an FN Mauser action. That gun would be near perfect, but it would be really hard to build a custom rifle like that today and stay under the $1000 price limit.
I would hardly feel handicapped with a Winchester 70 or Ruger rifle.....or a commercial Mauser.
