Autoloader? If that's what turns you on, go for it. I prefer to hunt with my Ruger #1 single shots. Caliber? Whatever, up to a point. I've shot a lot of Pronghorns and deer, including the largest, with my 6 mm Remington. I've shot Pronghorn, deer, and elk with my .280 Remington, functionally the same as the .270. IMHO the two most important factors in killing big game are bullet performance and shot location. First the bullet must be capable of reaching the vitals and doing sufficient damage. For over 30 years now I have used Nosler partition bullets in my handloads almost exclusively. I've recovered very few of them. One was in that very large Muley buck - he was over 200 yards away down a steep hill, facing away. The 6mm 100 gr partition bullet hit back of the left kidney and stopped just under the skin at the front right shoulder. Another time a different small Muley back facing me had the same type bullet take him mid chest and make it to the hind quarters. The last two elk I've shot, a nice 6x6 and an average cow, we both complete shoot-throughs on the chest with 140 gr 7mm partitions, both one-shot kills.
The other prime factor is shot location. I always try to shoot the animals in the chest.
I'd consider a 6 mm Remington/.243 Winchester marginal for elk, but they work great on deer/antelope. Any of the three you mention, with good bullets and shot location, are fine for deer/antelope/elk. Using an autoloader you may not be a reloader. Given that, I'd probably pick the .30-06 because of widespread availability of good ammo, but they'll all work fine.