Personally I like big rifles for big game. Of the 2 on elk, I'd want the .308, but if I was going elk hunting, I wouldn't be using a .308, I'd have my .340 weatherby, and my .300 win mag for backup.

For deer, I'd have the 7mm-08, especially for out west/ longer shots, the flatter trajectory and plenty of power to drop a deer.

.30-06 would be a better combo for deer/elk I feel than either of these two. Consider the .300 WSM as well, I wouldn't worry about kick out of either of these, and the .300 WSM is here to stay, so don't worry about it fading away from popularity.