I've owned more 7mm Rem Mags than any other rifle. It's a fine cartridge as are the others you mentioned. It's good for most any big gamein North America, though I'dwant something bigger for Grizzlies. Ammo should be readily available for this cartridge since it's very popular. For whitetail deer and such, a bullet in the 140 to 160 grain range should work fine. For larger animals like elk and moose I'd go with a bullet of at least 160 gr. and a premium one at that.

Between the 7mm mag and .30-06 it's almost a toss up. I don't believe either one has any significant advantage (the key word being "significant")over the other and I've used both. The .300 WM is a slight step up in performance, but at the expense of some more recoil. The .338 WM is a cartridge I have no experience with, so I'll leave the discussion of it up to someone else.

The bottom line is that in most big gamehunting situations you'll encounter in North America (except for maybe the biggest bears), the 7mm mag is a perfectly fine cartridge. Just make sure you're using a bullet that's correct for the game you're hunting and you'll be fine.