I'm not picking a fight over it as they are really so close. However I have a 26" barreled .243 and .25-06 and while I haven't shot Bergers in both I do shoot 115 Noslers, 117 Sierra and Hornady in the .257 and 105's in the Bergers. I've been able to get the 105 Bergers to 3100 fps and nearly the same with the 115-117's.
That said neither the .25-06 or .243 Win are ideal elk cartridges. Heck I've even posted that same video a few times, but that was a near perfect situation as you'll ever see for using a .243 Win on elk. Having killed several elk myself I can tell that I'd never recommend the use of a .243 Win or .25-06 for elk unless that was all you had to use. There are just far better options to use on elk regardless of what one video shows.