With the quality of 30 cal bullets availbale, I feel the 300 RUM is a good 1K rifle for elk and a fair rifle out to 1100--1200 yds depnding on bullet/load/ conditions etc. With a GS HV 177 it would be a good 1300 yd + elk killer.
IMO, there is no such thing as overkill. And at closer ranges the lower velocity of the 338 would probably be less destructive than the higher velociteis of the 300 depending on bullet and placement, etc.
The 338 EDGE (or RUM, or Lapua) of course will deliver a bigger payoad and make a bigger hole and reach just a little farther. All else being equal the 338 would be a more effective killer, but... the 300 will get the job done with good placement.
The 300 RUM is a cartridge that I can shoot without a brake by using a slip on recoil pad which can be removed for hunting if desired. I hate brakes. So, that is what prevents me from going with the 338. I might be able to shoot it with the pad, but I'm very happy with the 300
Speaking of elk.... it's time to go look for a couple of cows