I think that for most "average" shooters a fairly lightweight 30-06 (like a fwt or mountain rifle) is a bit more than they are comfortable shooting, even more so with the heavier bullet weights.
Given that, if I hunted primarily moose, in grizzly country, I'd probably be happier packing an 06 than most anything smaller on a 308 case.
I am a big 270 fan (flat trajectory, very effective on deer at longer ranges and very reasonable recoil level). The biggest animal I have taken with one is a black bear, or several big mule deer which were heavier than that bear. I would have no hesitation hunting moose with a 270 and decent bullets though.