Any of the three will do magnificently for any game in the lower 48, with the three species named in particular.
It doesn't take a cannon to kill any of them, even an elk - or archers would never get one. Any mid- to heavyweight bullet standard for any of those calibers will do fine.
Shoot 150-grainers in the 7/280s, 165s in the 308 and you'd never need to change anything for the game mentioned. One load, one sight-in, one full freezer.
Now it's all settled until Model 70 chimes in for the .270!
My choice? I made it 30 years ago: 308.