The 270 is a fine round for everything you mentioned with the exception of Grizz.
I lived and hunted in Alaska with a 7 mm rem mag. I was a kid and didn't know better.
Now I'm being told by all the experts that I was way under gunned.
30/06 is the standard. I'll take the 06 over the 308 even though I use the 308 on deer.
But not on Elk. I have a friend that has taken all his Elk with the 308 though.
Moose? I don't think so. But in Europe Moose are taken with the 6.5x55.
Go figure.
The 30/06 covers nearly the whole spectrum of big game in North America.
In Alaska it is the most commonly used round. As is the 300 Win Mag, and the 338 Win Mag.
You can find those three in just about any store up there.

Good Luck.

T