I'm no expert but
For that specific range of game tasks, if I had to pick one only, I'd probably run with either:
-- .35 Whelen
-- .338-'06 (semi-wildcat)
-- 9.3x62mm
-- .375 Win
-- .444 Marlin
-- .45-70 govt (or .450 marlin)
-- .338 WinMag (or other .338/.340 mag)
-- .376 Steyr
Roughly in order of preference, with the latter four being some overkill for whitetails and beating you up more, esp. the .338 winmag and .376 steyr. The others all also border on overkill, but when covering such a wide range with one gun, you have to go with the lowest common denominator, so to speak. The reason I put .45-70 and .444 ahead of 338 winmag and .376 is I think you'd have less meat damage on the smaller animals with those slower bullets. Heck, those 2 might be your best choices period, for this wide range of tasks, unless you need a lot of range/trajectory, like you might out West, because you're giving up some trajectory in them (.45/70 and .444). .375 Win also gives up a tad of trajectory, and it's not so common of a round, but it can be loaded quite heavy and would have pretty mild recoil - seems like a great brush gun round. There are other .35s, plus or minus, to look at, like .350 Remmag, .358 Norma mag, etc. You might even go a little smaller to a .323, like 8mm Mauser (aka 8x57JS). That's a great all-purpose round as well.
.30-'06 Spfld seems a little light to me for brown bears -- for black bears, meese, elk, caribou, buffalo, yak-type critters, and even smallish inland grizzlies possibly, but the coastal browns (kodiaks) can be humongous and hacked off, and I'm assuming you wanted to include them as a possibility - so I'd want .338-'06 minimum with heavy bullets if there's any chance of close contact. .338-06 is not really ridiculous overkill on small deer either, and has some decent range, so it would be a nice compromise. Ditto on the .35 Whelen and it will be easier to find on gun store shelves. Ditto 9.3.62mm (I think that's a .366 diameter round). Plus, if you went with one of the bigger boys, you could add polar bears to your list, and have then entire continent covered with one gun. Now, if you eliminated coastal browns and polar bears from the mix, I'd say .270 win, .280 rem, .308 win, .30-'06, will do it all, or a 7mmRemmag or .300 winmag if you need to really reach out there some distance. Like someone said, a case can be made for just about any cartridge. But, OTOH, what do I know - like the other gentleman above, I'm mostly an armchair hunter, esp. when it comes to tuque-land (canadian/alaskan) hunting, so far. Mr. Eatman and others have been there, done that, so I'd listen to them more than me.
P.S. I'm surprised that I mentioned the .444Marlin, but "444", you didn't.
And the thought of a .375 H&H mag or other large belted magnums being used against a poor little woods-grown bambi tickles my funnybone, though I'm sure they'd work. .375 HH mag may be the quintessential worldwide all-purpose round, but perhaps the .35 Whelen or .338 winmag is the quintessential north american all-purpose round (arguably). Now, like someone said, if you're going to roll your own ammo, then something pretty powerful like the .338 winmag starts to look a lot better as the most versatile round, because then you could upload or download as necessary.