" I particularly like the 7x57 Mauser (If you can find them, Norma made 175 gr steel jacket bear loads)=light recoil, light rifles, good accuracy and excellent killing power=after all, Karamojo Bell killed over 100 elephants with one!"
My thoughts exactly. A handloader can make that cartridge snap at the heels of a .270 or .280 Rem. with ease.
However, as the cartridges in question are the .270, .308 or 30-06 I'll just add this. Col. Townsend Whelen made this comment, "The 30-06 is never a mistake." I concur 100%. Interesting point though is a few years back I did a little test using some factory ammo for the .308 Win. and 30-06. Rifles involved were a Winchester Youth Ranger in .308 Win. and a J.C. Higgins M50 in 30-06. Both rifles had 22" barrels. Ammo was Winchester 180 gr. Power Points. Average velocity from the .308 was 30 FPS FASTER than the 30-06. Also, the 30-06 did not reach the 2700 FPS advertised. I wasn't all that surprised and ammo is loaded to a MAP or Maximum Allowed Pressure. Velocity is close up to a point but will vary from rifle to rifle. The point being you may or my not get the velocity you think you're getting. No matter, it'll be close enough.
I've hunted deer and elk for over 60 years. Killed my first deer at age 11 and am now 84. I've taken deer with everything from a .22LR to a 30-06 although I have hunted them with a .375 H&H. Just didn't see any deer that year.
I've hunted elk with the 375, 30-06, 270 Win.,280 Rem. 300 Win. Mag. and my all time favorite the .35 Whelen using the 225 gr. TSX bullet at 2710 FPS. Trajectory is virtually the same and my 180 gr. handload for the 30-06.
Longest shot made on deer, one at 426 paces with a .308. Several 300 to 350 yard shots with the 30-06. One elk at 530 yards. I knew the area so took a .300 Win. mag. and did over a period of three month practice at the range. The shot was easy at a laser ranged distance of 530 yards. As I said, I knew the area. Once a shot was fired on opening day the elk would go into these several mile wide "parks" where they could see coming from a long way off and walk away. Longest shot with the .35 Whelen FWIW was 350 yards.
I haven't used the .270 very much. It just doesn't move me. I've shot a few deer with it and they just died. I set it up for distance for my one and only guided antelope hunt and when the shot came it was all the way out at 75 yards. I could have taken that shot with my Great grandfather's old 30-30 Winchester carbine and a cast lead bullet.
So, which would be my choice of the three the OP was looking at? The .308 or 30-06 would be a tie, then the .270. Like the Col. Whelen once said, "The 30-06 is never a mistake,"
Paul B.