A 7 mag and 308 will both kill elk, but there are distinct differences, hardly just a matter of flipping a coin. If you choose the rifle carefully a 308 can be 2-3 lbs lighter making it easier too carry around at 8000' after elk. Buying a heavy 308 for elk hunting makes no sense.
With the best loads a 308 starts running out of gas on elk size game at around 400 yards. If you don't have the skills to take longer shots, or if weight reduction is primary a 308 is a good option.
A 7 mag will have noticeably more recoil, but to be honest it is tolerable to most people. Serious recoil starts with the 300 mags. But at 400 yards it has almost 500 ft lbs more energy and will drop more than 13" less compared to 308. You still have enough power to take elk at ranges far past 500 yards.
Not a 25-06 fan even for deer. Yes it'll kill em, but a 25-06 won't do anything a 243 won't do just as well. It can kill elk, but would be the bare minimum.
Rounds like 270, 280, 308, 7-08, 6.5, 30-06, any of the various 300 and 7mm Mags or even the 338 mags all kill elk. Heck a 30-30 will kill elk. It is just a matter of how far you can and want to take shots and how much recoil you are willing to tolerate. How much weight you are willing to lug around at high elevation in rugged country is a factor. Everything is a compromise.
For my money a 30-06 is the all around leader although a 270 or 280 would do nicely as well. Enough punch for 500+ yard shots and manageable recoil with reasonable light rifles. If someone really wants to cut weight and is willing to limit shots to around 400 yards the 308, 6.5 Creedmoor, or 7-08 make sense. If you want to truly go long range then the 7mm, 300, and 338 mags are your huckleberry. At the cost of heavier rifles, longer barrels, and more recoil.