I think most everybody is right.
The 8 mm is overlooked by many,but it is as serious as a 30-06 and can be used to get the job done.
But,I also consider,I have a .257 AI.If it was the only rifle I had,I would hunt elk with it.I'd use 120 gr Nosler partitions and be inside 200 yds.I would only ask that bullet to penetrate ribs to get to a heart lung shot.Nothing more.I'd pass up any shot that seemed risky.
I do not take the .257 because I have another rifle more suitable.
I do not know your circumstances,but if I were going brown/griz hunting or berry picking,I think I would like a .375. But I would also consider an 8x57 a rifle I could use,just as the .257 on elk.
I recall mentioning in one of the "Is my 9mm with 124 gr HP's a good bear defense?" threads that an old beater non collectible dark bore pitted exterior
Mauser 8x57 chopped to a barebones carry carbine would be much better than a handgun.
And,actually,up close,a .303 Brit,30-40 Krag,or your 8mm may perform better than a .300 WSM. Long,penetrating bullets at moderate velocity are dependable.
I'm not a bear hunter or great game expert.This is my line of thinking
Just remeber a brown bear with a perfectly placed heart shot is mortally wounded and he will die.It might take him 20 seconds to figure out he is dead,and he might bite your head in that time.
If,however,you bust him where the shoulders and spine come together with a big,dependable bullet,he is more immobilized while he is dying.
Breaking big bones is the reason for big guns.