For the species you mentioned, the 30.06 will do everything you want with less recoil. You can use 150 or 165 grain bullets for any of those animals. I'd probably go with 180 grain for Elk though. If a Moose get's in the way, I'd go with the heavier 200 grain bullets. The 30.06 will normally have a 22" barrel while the 7mm (depending on make) will run 24-26".

If you decide to reload later, you can really have fun with the .06. Like 110 grain bullets for shooting prairie dogs and 'yotes.

I would never feel undergunned with a 30.06 on my shoulder. Unless I was facing a Grizzly or other dangerous game.