With similar bullet weights a 30-06 and 7mm rem mag are nearly identical out to 300 yards--the maximum distance MOST hunters should ever attempt an ethical shot at deer-sized animals. The most important thing about hunting is being able to hit your target. A 22-250 would do the job just fine with good bullet placement (along with the proper bullet). It's not what you've got, it's how you use it......
I use a 7mm mag and a .300 win mag on our small deer in coastal California and with the right bullet I get excellent performance and minimal damage. Shooting a deer through the shoulders is a guarantee of bloodshot meat IMO; through the chest behind a shoulder puts 'em down with little meat loss. I agree that true hunting is getting in close; there's NOTHING like shooting a good buck at 10 yards with a bow, or stalking in and waiting on a harem'd up buck and plugging him with a muzzleloader.
I would say you have the right idea to practice with your equipment. It shows respect for the animal you are hunting. Either rifle will work just fine; pick the one you shoot best and feel most confident with. Only you can judge your personal hunting conditions.