The 308 caliber is another short cartridge option. The 7mm08 and 243 are necked down versions of the 308 after all.
Trust me, if all you want to hunt is everything under the size of an elk, all three of those calibers, as well as what you listed, are perfectly capable of taking your game.
My father started out using a 30-06 after he got out of the military in the early 70s. Yeah, for a change he used a .44 mag rifle, but the 30-06 was his primary rifle. He took many deer with the gun, and then moved to a .270 about 15 years ago. He made the change from his heavy semi-auto 30-06 to a lighter bolt action .270 and took many deer with his A-Bolt.
Then, because he wanted a lighter kicking gun, and new backup rifle, he bought a .243 that he made his primary gun last year. The buck he shot last year was just as dead with a shot from the .243 as with a round from the other two rifles he's used.
If you are not worried about bigger game, then you can choose from such a wide variety of calibers that are honestly equally good for taking most North American game it's not even funny.
Of the three calibers you are considering, I'd go .270 if you don't reload your own ammo. If you reload then go with the .280. The one nice thing about some calibers is that if you forget/lose your ammo then you can to any store that sells ammunition and be back in the woods in relatively no time. Which is a downside of the .280 at this time being not as popular as some other cartridges.