I think with Elk it is not so much about the caliber of the bullet, 257, 264, 270, 284, 308, 311, 318, 323, 338, .... You get the picture.
The different cases are going to determine how fast you can send that bullet. (how Much powder and pressure you can use based on your action)
With Elk I think the better idea is how far away you can be and shoot the elk.
308 Based cases are 200 - 250 yards (243, 260, 7-08, 308, 338 Fed, and 358 Win)
30-06 based cases are 250 -300 yards (25-06, 270, 280, 30-06, 338-06, 35 Wh)
Win Belted Mag case 300-400 yards (257 Wby, 264 WM, 7mm Rem Mag, 300 WM, 338 WM)
And the hot mags 400 - 600 yards (7mm Wby, 7mm RUM, 300 Wby, 300 RUM, 338 Wby, 338 RUM,.... You get the picture)
These are the ranges where the standard projectiles will carry enough energy to penitrate and open. Can you shoot farther, probably.
Heck the world record elk is held by a guy who used 30-40 Krag in the same league as 308.
Many Elk have been taken with 30-30, 32 Win Spl, 44-40, 44 mag. The 44 mag that is good out of a pistol is no where near the horse power of the 30-30.
The 7-08 is to the 308 what the 280 is to the 30-06. Same case necked from 308 to 284. .024 of an inch differance. Or about 8% the width. If you only have a 7-08 and have the oppertunity to go elk hunting I would go. I would just limit your shots to the 200- 250 yards.