70 years ago, 300 Savage was widely recommended for ALL North American big game. This cartridge is similar to .308 and 30-06 out to about 175 yards. But beyond this distance, the larger cases hit harder due to increased powder capacity and greater velocity.
Many of the colossal bears have been taken with old time cartridges such as 45-70 and others.
I'm thinking that .308 & 30-06 loaded with a 200 grain bullets of bonded core design should do well if the hunter does his part. That is, wait for a broadside shot and place the bullet(s) through the shoulders. That being said, I honestly admit that I've never hunted the great bears. Largest animal I ever killed was a Dakota bison and it folded up on the spot from a brain shot at approx 125 yards. I've also taken one Canadian moose with my .308 - two chests shots did the job nicely.