I think it's all relative. A 1" rifle will kill prairie dogs out to as far as you want to shoot them, just like a 1/2" rifle will. But the 1" rifle won't kill them as consistantly as the 1/2" rifle will. It just depends on how serious you are. If you're the kind of guy that counts kills, and calculates your kill percentage on your lap top at the end of the day, you probably need a $2500.00 dollar rig that shoots 1" groups at 1000 yards
If you're just out there to shoot and have fun, shoot what you've got. I've shot these things all my life. I started with a lever action Marlin .22 with open sights. And I killed a LOT of prairie dogs. Use what you've got, but know HOW to use what you've got.
As far as the .223 being out of gas after 300 yards........
I shot a .223 a lot longer than I'm willing to brag about when I was in the military (with open sights at that). It will do the job on coyotes out past four hundred yards, I know because I've done it. It just doesn't kill them with quite as much authority as something like a 220 Swift. As long as you know what you're doing, it will do the job waaaaay out there.