I appears that the rancher in the story is not hunting grizzlys. He is merely killing them as pests.
I would not use a .243, myself, as I do not think it would have enough penetration at a number of angles.
However, I remember a conversation I and my brother were having about “elk” rifles. My 80+ year old uncle, who was an absolutely superb woodsman, a professional trapper all his life (until 93) who had shot lots of elk, spoke up and said “a .243 is plenty for elk”.
It makes no difference why he wants to kill them it takes the same amount of energy to kill them either way. A .243 will kill an elk. I have a rancher friend in Idaho who kills a nice elk with a 22-250 every year. He sits in his blind and head shoots them when they come down to feed on his ranch. It is not often that you can create the situation where you have a good rest and a clear close head shot on an elk. It works for him but would not work up the mountain in the trees and brush. One more point, elk almost never attack when wounded or flushed out at close quarters big bear do.