None whatsoever, do you think there's any difference between a bullet made with the exact same materials and technology that's .007" (seven thousandths of an inch) bigger than the .270's you're considering?

They're identical in use, the .270 and the 7mm, the 7mm just has a lot more projectile options.
I'd look farther from your current range, as the cartridges you've mentioned really won't do anything your four .30-06's won't already. The mag may add a touch of reach, but 99% of hunters can't and don't use the range and buy it under curious pretences. A .30-06 can do 400 yards no problem for a good hunter, and who takes their game further than that, really?
Why not look at something more coyote to deer range, like a .257 Roberts, and something more moose, elk and up range, like a .375? Then you'd add capability to your cabinet, not just more choices to make with a blindfold on!
Good luck!