ELR as defined by myself, for me, is 1500 yards and beyond.
Why that distance? I hunt/shoot ELR from a hide. My definition of a hide is a nice comfy spot with great terrain, comfortable prone shooting spot with great all around visibility. Most importantly located near elk habitat.
In other words, the hide is pretty much like my basement floor for solid shooting position. I hunt/shoot from these hides year round. I know what I'm doing there. I'm accustomed to the wind at various times of the year.
Having said all that, caliber/cartridge is not as important as being able to confidently place the shot. It's all about consistent accuracy, bc, time of flight, impact velocity/energy and bullet terminal performance.
It goes without saying that as distance increases the heavier the bullet the better. The 338 caliber in a Berger 300 grain offering, in my estimation is minimum for ELR.
Cartridge wise, I feel the 338 RUM is a bit on the small side. Too much time of flight and too high mid range trajectory. The Lapua and Edge is only a bit better. The Terminator, as in Broz's rig, is solidly in the great choice area. The really big ones, 408 based top that. However, bulk, shoot ability and portability come into play with those big heavy suckers as I can attest from messing with ma "Big Girl".
As JE says "Unless you intend to shoot over 1500 yards all will do the job." I'll add "if you can reliably place your shot."
Oh, and ELR shooting has more to do with the shooter than the rig being shot. That is assuming all rigs are created equal.