Somehow I just wiped out my own long post. so I'll just say this. If I had a 7-08 and didn't like recoil, I don't, I would not hesitate to use it elk hunting. I would set it up to shoot a 6" target at MPBR with a 160gr bullet and live with it. You would be more than adequately armed. I would use the 162gr Hornady Inter Lock becaues I don't like spending money on premium bullet's that I don't believe are needed. I have shot elk with 7x57's and the 154gr Hornady RN, nom problem. You do not need a cartridge that will work to 300yds. There is nobody that needs to shoot that far; there are people that chose to shoot that far. MPBR at a 6" target with a 162gr bullet from a 7-08 will allow you to hold dead on to 266yds! Well actually even farther than that and the most important element here is bullet placement assuming your using an adequate bullet. The 160gr class of bullet's in 7mm are more than adequate, cup and core up to TSX type.
The effective range will be limited more by your ability that the cartridge's.