I'm a big fan of the .243 cartridge. My brother and I plus our families have toppled well over 200 antelope and close to 100 mulies with this cartridge. Winchester designed it as a varmint / deer cartridge and within these parameters it does its BEST work.
Several states allow the use of .243 and 6mm for elk hunting. It's not
illegal in Wyoming to hunt elk with the .243 although this would not be my choice.
This large bull weighs about 700 - 775 lbs in my estimation. The angle is perfect for a double lung shot through the ribs. But most bulls do not present such a shot and many hunters lack the skills to get close for a sure shot.
I've met many good hearted Easterners over the years. 30-06 seems to be the most common rifle for their "western big game hunt" and there is very good wisdom with this choice.
Good hunting to you.