In my experience cartridges around the .243 Winchester in work fine, but so will .224 centerfires from the .223 on up. I knew a Montana game warden years ago, when the MT Fish, Wildlife & Parks Department used to kill "problem" elk off rancher's haystacks. He preferred the .220 Swift, whether for head or chest shots. He killed hundreds.
Of course, it also depends on hunting conditions. If you're hunting in open country where specific shot placement/angle is more easily acquired, then smaller rounds definitely work. One of my local friends is a retired outfitter who only hunts cows anymore, and he prefers .224 rounds like the .22-250 and .220 Swift, because he doesn't want the bullet to exit--as it did most of the time when he carried a 7mm Remington Magnum while guiding.
Back when I started elk hunting here in Montana in the 1960s, most public-land elk spent much of the rifle season in pretty thick timber, where shots were rarely as long as 100 yards. Consequently most hunters considered the .30-06 with 180s as minimum, but I eventually went to 200-grain Partitions, which proved adequate for any shot angle.
But another factor, even back then, is where I hunted most was close enough to Yellowstone Park to run into grizzlies--partly because that was when the Park had just closed down the open dumps and bears were wandering farther to find food. Never ran into one there, just tracks--but have run into them while elk hunting in other parts of the state, and while pheasant hunting in northwestern Montana on the Flathead Reservation, where several hunters have been mauled in the past few decades. And they keep "expanding." The area where we live didn't have any when we moved here in 1990, but they're found in the both the mountain ranges on either side of the valley--and it won't be long before they're in the valley itself.
Grizzlies are why I tend to prefer something a "little" heavier when hunting in their territory. Twenty years ago I got a bull elk and moose on a horseback hunt in northern British Columbia, in an area where my guide and I ran into 10 grizzlies in nine days of hunting--one of which tried to come in while we were taking apart my bull moose. Luckily, the horses spooked after scenting the bear, warning us, and we managed to get the job done without a problem--though we kept my .300 Winchester Magnum and 9.3x62 handy while taking turns during the dismantling. When we rode out a couple days later, on the main trail that went right past that spot, a big boar was sitting next to what remained of the carcass, his stomach bloated by moose parts.
On the 9th day of that hunt I killed a 6x6 elk in thick riverbottom cover, near where one of the guides in camp had killed a charging grizzly a couple weeks earlier. I used the .300 Winchester on the bull, at the vast range of 75 yards, and didn't feel I was "overgunned" in those circumstances....
