Kindof a fun-gun/SHTF sniper rifle. And of course, if I choose the .270, I'll use it to hunt coyotes too.
The 270 is not an ideal coyote rifle if you want to keep the fur, and if you don't any of the three are equally good.
I doubt you could tell a difference in the rounds by what they do to a coyote. A whitetail deer, either, for that matter.
You might notice a statistical difference on an elk, if you shot enough of them at longer ranges than 300 yards.
A lot of these discussions are angels on the head of a pin. The recoil of a 270 and 308 are near identical, and the 06 is just a little more.
As to flatness of trajectory, you're talking about 6 inches holdover at 300 yards as compared to 9 inches, and maybe three or four inches more or less at 400 yards.