My black bear hunting exploits mostly took place in the 1970's. I used a .303, .30-06, and a .350 Remington during that time. Of the 3, the .30-06 was so much better than the other two that it wasn't even a contest.
The best .303 bullet was Remington's 180 gr round nosed Corelokt. I never saw this bullet fail, and in my rifle, it was accurate enough with the peep sight. I only used factory ammo in that rifle, and it is the only centerfire rifle I've owned that is true of.
The .350 was loaded with 52 grs of 3031 which pushed a 250 gr Hornady close to 2500 fps, and while powerful, it was not outstandingly accurate. With todays bullets, better powders, and chronographs to maximize it's potential, I'm sure the .350 would be a top performer but, it was a bit of a disappointment in the early '70's.
I was able to shoot that .30-06 more accurately than any other rifle I owned up until that time. It seemed to kill as well as the fat little .35, and reach further. I fell iin love with the 165 gr Nosler and Hornady bullets in that rifle, as these could reach along lake shores and down power-lines in the bush.
Jump to present day.
The .30-06 remains a favorite. Loaded with modern bonded core and mono metal bullets the '06 is suitable for any North American game, and perhaps is suitable for any game in the world other than thick skinned African stuff. When loaded with the old standbys - Remington Corelokt bulk bullets and Hornady Interlocks, the '06 remains a very capable deer and black bear cartridge. My greatest successes and worst failures have been with the '06.
The .375 Ultra might be considered a bit over powered for blackies, but so what. This rifle does everything so well. If were half as accurate as it is, it would still be twice as accurate as it needs to be. It shoots as flat as a .30-06, and it's excessive power does not cause excessive damage to light game. When I reach for a hunting rifle today, it's this one.