For elk, .280 remington, definitely. Maybe remmag if you're a masochist.
It's heavy to carry, but the recoil ain't that bad on my rem mag, no worse than a .30-06 and it's got a little flatter trajectory than the 06. I really like the rem mag. I've found the 160 grain Federal Premium ammo with the 160 grain Nosler Partition a very accurate load in my Savage. I can't handload the caliber much better than this factory load. It'll take an Elk about as far away as I'll ever shoot at one. The STW is excessive. So what it's flatter shooting? I ain't shootin' at game at 500 yards, I'll get closer, thanks. To many variables to risk a wounded animal at long ranges. I'll keep my shots preferably under 200 yards, but might stretch it on a big bull at 300-350 if the wind is calm and I have a good, solid rest and I ain't shootin' up or down hill too much. IOW, the shooting conditions have to be ideal before I'll go much over 200 yards for a shot, so I don't see the STW as advantageous to me.
I've owned a 7x57 before in an old military surplus gun. A light hunting rifle in this caliber would be sweet to shoot and powerful enough at all, but excessive ranges for Elk. The 7 mag puts down a lot more energy on target, but the Mauser round has enough and enough bullet to go with it. It's a better round for the heavies than the 7-08 IMHO.
The three I'd pick would range from 7x57 to .280 Remington to the Rem Mag. When I bought my Rem Mag, I was thinking about a .280, but figured, heck, if I don't like the recoil, I can always load it down, can't load the .280 UP to match the big magnum. And, they're both only available in long actions of 06 length. My Savage ain't pretty, but it sure does shoot! Pretty is as pretty does. And, it ain't the cannon at the butt end you'd think it would be. Heck, though, I'm used to poppin' half a box of 3" 12 gauge loads in a morning on Ducks. Compared to that, the rem mag is a .223. LOL! You feel it off the bench, but I have a Past pad for that. Out hunting, it's not bad at all. My biggest buck was taken with that rifle, though it ain't exactly a record book buck. He was quartering away at about 150 yards and I slipped the 150 grains Sierra Game King perfectly behind his right shoulder off my shooting sticks. He dropped where he stood.