When I was first gearing up to begin my Hunting journey in 2014, I read a lot about this same topic.
While obviously a .243 could do it, especially with a copper round like a Barnes TSX. Something somebody said on one of the responses made a lot of sense to me.
He mentioned how when bears get shot, they tend to want to run into the thickest, most dense cover they can find, and the guy was basically saying that having more energy on tap can help tip the scales in your favor is terms of the bear going down right there, or in doing less running, equating to less tracking in tight thick stuff that'll make ya nervous as hell trying to follow in after it.
Same guy also said to give 'em a good amount of time to expire. No sense in risking life and limb. And Ooohweee.. this one dudes pictorial he showed of how he had to track the bear!... Oh man... talk about pucker-factor! Dude had to crawl in under brush...listening carefully... moving forward slowly... until he finally spotted the bear... and that dudes bear... it wasn't completely down for the count yet. It was grrr'n at him, though obviously close to death... deep within that pucker brush, and way closer than any of us would like to be from a bear that's not 100% dead yet. I think he said 15yds or so, something like that.
And it was that thought... that swayed me towards opting for a heavier caliber instead. Especially being newer, figured more energy the better, even though I don't care for the recoil from it. While actually hunting, it doesn't matter... you're gonna be so pumped you won't even feel it. My first deer I was leaned-in against a large rock at a weird steep angle and got clocked a lil bit with some scope bite. I knew it had knocked against me, but I didn't feel significant in that moment. But later on the next day when I woke-up had a nice little nickel-sized shiner in the corner of my eye .