The 7mm-08 is a great cartridge and should not be underestimated. All the same, a bull elk, at 700+ pounds, is one seriously strong animal. Badly hit by anything they can and will leave the area, and leave the hunter trying to keep up...
Some elk hunting is at relatively close ranges, and a short-action cartridge is fine. Many times though, it's one of those famous "cross-canyon" shots where the extra velocity of the magnum comes in handy. Elk have been killed with all sorts of cartridges, from the little .243 Winchester up. When I'm hunting them, I like to treat them with respect, and carry enough rifle. The 7mm-08 you're considering will do the job, but a 7mm Rem mag, or the STW, will do the same job at greater distance thanks to greater velocity.
If you want the 7-08, get it - you'll likely have a superbly accurate rifle that will excel as a deer slayer, and will work, within certain limits, as an elk cartridge. Do yourself a favor, and work up a good load, with really good bullets before you zap a 700 pound bull elk though.
Cartridge choice is nowhere near as important as bullet placement and bullet construction. At least not in my ever so humble opinion!

I'd have no reservation about taking my little old .308 for elk - but I'd make sure I had a real good load, with deep penetrating bullets. Something like a Nosler Partition or a Barnes TSX. In 7mm, the 175's do an amazing job of penetrating...
Regards, Guy