My .260 ai, which is not a huge jump from a 6.5 cm, has taken 5 elk from the closest at 703, to the farthest at 940, using a 140 Berger vld-h @3040, 147 eld-m@3040, and a 156 EOL @2930.
Elk 1 was a 6x6 bull, center shoulder hit at 703 with a 140 Berger vld-h. Went in between scapula and humorous, exited same place, elk lunged forward and spun around at impact, stood wobbly for 8 seconds, and tipped over by 10 second from impact. Have it on youtube.
Elk 2 was at 816, 5x6 bull, 147 eld-m. First shot was back, double lung and liver. Elk ran 15 yards and stopped, I reloaded, held off, and shanked my second shot, pulled low, put the bullet through some skin on the leg. Reloaded, fired again. Center punched the shoulder. From first impact until elk was down was 34 seconds, time from second hit until elk was down was 15 seconds. Have it on youtube.
Elk 3 was at 703 yards, my wife's, a large cow, 147 eld-m. Elk was slightly quartered to us. Impact was through the elbow, at the angle it should have taken out both lungs, and exited about 6" behind the off side shoulder. Bullet deflected, only hit a portion of the near side lung, went back into the guts, missed the liver. Cow ran about a mile, about 600 yards as the crow flies. Found her dead, bullet was under the hide in the rear off side quarter. Have video of the shot on YouTube and photos of entry, showing the angle and impact, making it obvious that the bullet deflected significantly. Stopped using the 147 after this.
Elk 4 was at 880, large cow, 156 EOL. Shot was high shoulder, double lung behind the shoulder, under the spine. Cow dropped like she was struck by lightning. Have it on video.
Elk 5 was at 940, medium cow, 156 EOL. Shot was slightly quartering to, took out the top of the heart. Cow ran 10 yards and dumped. Have it on video.
It can absolutely be done effectively and efficiently, however bullet choice, impact location, and other factors of the shot must be taken into account more and more as you go to a smaller projectile. That is not an excuse to get complacent with a .338, but it's just as ridiculous to say a 6.5 is not adequate for elk. Think of your hunting style, ranges you shoot, and much more importantly, your abilities, and make the proper choice in cartridge selection, bullet selection, and distance you are proficient at.
My guess on your lost elk is either poor shot placement, or a bullet failure as I saw with the 147 eld-m, or both. It's highly unlikely you made a good shot through the shoulder and both lungs and the elk just ran off. A low hit, i.e. heart level, but back a few inches behind the heart, makes it very possible for an elk to run a long ways, leave a significant blood trail in snow, and not die for a long time, if at all. This is my guess as to what happened. However, as no elk was recovered, it is just that, a guess, and means about as much as the dirt on the bottom of your shoe.
As for me personally, after killing many elk from bad breath distance out to 1036 yards, I look at a magnum 7mm with a heavy c&c or medium to heavy mono as my minimum for an elk rifle for my style of hunting, with preference going to .30 cal and up. For the next gent, a .243 with an appropriate bullet could be perfectly adequate for their hunting method.