RE: Big Game Hunting
You're absolutely correct usa . . . Most would think, "Bigger the bullet, bigger the damage". Not so. You're hitting a small bodied animal (deer) and it can only open so much.
Crawfish . . . To answer your question. Don't take a .270 to Alaska to shoot a Griz! You'll just piss him off. You said ALL North American big game. That's all deers, bears, elk, moose, antelope, cats, goats,
sheep . . . you get the picture. I'd have to vote for a .300 Win Mag. Big enough for Griz, moose, elk . . . and still great for the antelope, deer sized animals. Cats and goats aren't real big either. The balistics are indisputable. Flat shooting to 300 yards+. Which you'll need for some. Lots of variation in over-the-counter ammo and likely to be in every shop in North America.
$1,000.00 budget? . . . I'd have to go with the Remington model 700 in .300 WinMag($665) and a Nikon Buckmaster scope in 4.5-14X40 with the BDC reticle ($300). You already have slings, cases and stuff.
The Nikon scope blows away the Leupolds for clarity and low light gathering. I A/B'd them at Cabela's and it blew my mind how much better the Nikon was than Leupold. 4.5 is as low as you'd want to go and 14 power is very handy for long range shots on lopes and sheep/etc.
My (more than) .02 worth.