The .270 will have less recoil since typical bullets used are 130-150 gr whereas .308 and .30-06 typically have 150-180 gr.
You won't see much difference in trajectories inside 300 yds. But, it will be much easier to find commercial Match ammo in .308 than the others. You can find Match .30-06 but less variety, and never heard of commercial Match ammo fir .270.
All three will do anything you need with the correct bullet up to black bear and elk within 300 yds. All three should be readily available on ammo shelves but check your local stores to be sure and compare variety of each.
I personally like the .30-06 because of my M1 Garand and hand me down hunting rifle. But rationally I would go with .308 because it is the most popular nowadays and will have the greatest availability of ammo variety: Match, reduced loads, FMJ bulk, varmint, Wolf, etc. if a manufacturer sells a special bullet or ammo the first caliber it appears in is .308.
If you want to minimize recoil and don't plan to use anything but regular hunting ammo then the .270 would get the nod.
Consider the .308 as the default choice and ask yourself if either of the other two offer something worth giving up the widest choice of ammo on the .308. I am assuming you do not reload.