I started with a 308 and that was OK then and probably still is - I doubt that elk are tougher now. But I was young and climbing up and down mountains didn't bother me as much as it does now. I have never felt the recoil of a rifle when I shot at big game - so a 338 is as easy to shoot for me as a 7x57. I am now too over-weight, I have bad knees, and I'm afraid that if I hauled an elk up a steep mountain side it would take more than a day and I might end up in E.R.
Probably the choice of the bullet is more important than the caliber all the way from the 260 to a 375.
I started using Nosler partitions when they were made on screw machines. I still believe that a 260 Nosler partition bullet is better than a 30 caliber Sierra bullet designed for something other than Elk. Nosler partitions never let me down.
Using a varmint caliber though...

For the time is coming when people will not put up with sound doctrine, but having itching ears, they will accumulate for themselves teachers to suit their own desires, and will turn away from listening to the truth and wander away to myths.