Shooting a Grizzly with a 270 is about like shooting a big northern deer with a 243 or a small texas deer with a 223 scale wise. Not a problem, just hit them right with a good bullet. A 338 or 375 isn't going to kill them any quicker if you if you don't hit them right.
Throughout time I'd hasten to bet more grizzlies have been killed with 30-30's and 30-06's than anything else, add the .303 for Canada.
Eskimos like the 223 and 22-250 for polar bears.
Shoot it with whatever blows your skirt up.
Saw one shot 9 times with a 358 norma mag and the thing was still flopping around. Saw a 30-06 180 hornady interlock bang flop one with one shot. The 30-06 bear was much larger. Shot placement is what matters.
