While there is no arguement that it has, and continues to be used for elk and bear, it would not be my first choice for either.
Make your shots count, keep the range reasonable and it will take bear reliably. I have no experience on elk, but I understand it has been used on moose. The result was in an article in "Ontario Out of Doors" Took some seven shots to put down the moose.
No doubt, using premium bullets at fairly close range, and picking your shots, one shot kills are possible.
I'm telling you to get a bigger gun for anything bigger that black bear for saftey, and for respect of the animal.
As to it being more effective that medium bores on game like deer and bigger:
I have seen it drop deer like lightning from a nice well placed shot. But nothing I have not done with the 30-30. In fact, the two are more compareable than many would admit. The 243 is flatter shooting, and has longer range on light game like deer, it has an affinity for premium bullets. The 30-30 carries more bullet mass, and does not require a premium bullet for penetration. In a bolt gun, it can be loaded with a pointed bullet, even a boattail, and have range, and trajectory compareable to the 243, with more energy.
Comparing it to anything larger than a 30-30 will be an uphill battle for you, unless you are varmint shooting.