Scorch hit it 100% right here;
Most moose hunters carry 300 or 338 mags because there are usually bear in the areas you kill moose.
I have killed 4 moose in my life. One, killed with a 62 cal flintlock dropped at the shot. The rest seems unimpressed with my rifles, (375H&H and 348 Winchester) but they don't seem to get scared and run like an elk will. They just went about their business after my shots ----and then fell over.
I loaded some ammo for a hunter a few years ago for her 270 winchester. One shot at her moose and it stood there for about 5 seconds, tried to walk and only made it about 1 body length then fell.
I used to go to church with a man that killed his moose with a 243, one shot through the heart. Again, the moose seems a but underwhelmed and decided to walk away. But then it fell dead.
Big bears however are a different story.
In many of the areas I hunt here in Wyoming we have a LOT of grizzlies, and so I have a tendency to carry a powerful gun when hunting elk or moose (when I can get a license) not because I need the power for my game, but because I feel better with a big hammer in my hands when I know I may not be at the top of the food chain.
I had hoped to hunt elk with my 6.5X54 Mann/Scho this year, but I drew a license as a 2nd choice area, and that area is full of bears, so I will be taking my 9.3X74R, my 375H&H or maybe my 300H&H. My 1st choice has a lot of wolves, but not many big bears.
I will still try the old 6.5 on deer however.
One last note that needs to be pointed out.
If you are carrying a big rifle for the use on a bear in an emergency, it's a waist of effort and folly to use ammo made for the elk or deer. About 12 miles from Dubois Wyoming, about 20 years ago there was a very bad mauling that was in all the national outdoors magazines. It happened about 300 yards from where I was at the time and I heard the screams and the shot. I big sow attacked a man and had him down, ripping him up bad and pulled his are out of the socket. his partner ran up to the bear and at powder burn distance shot the bear with a 300 Win mag. Wrong bullet! It was a 165 gr. Sierra loaded in Federal Premium ammo as I recall. The bullet hit the bear in the shoulder and didn't do the job. thankfully the bear was diverted from the mauling and had enough so it ran off. US fish and Wildlife and Wyoming game and fish as well as Fremont County Sheriffs all converged on the seen. Even with dogs a 2 day search turned up nothing of the wounded bear. Game and fish was going around to hunters and camps and telling them there was a wounded grizzle near. It was tense for a while. They never found the bear and we all hoped it succumbed to the wound and died. But before it did it made it over 1.5 miles and the dogs lost the sent at a few of the mountain creeks.
The point?
If you carry a big gun for bears when you are hunting elk, load with bullets for the bears! Not the elk. Bear bullets are those that can get through heavy muscle and bone at close range and not blow up. They will still kill your elk just fine. Don't fall for the fad of needing some "VLD long range" wizz-bang uber slick arrow" bullet. Even in open country you almost have to try to get shots over 400 yards, and if you can hunt at all, you will kill 99.5% of your elk an any country at 300 and closer. It's unwise to prepare for the 1 shot in 1000 instead of the 999 shots out of 1000.
As a "go to" ammo for elk and moose where there are grizzlies, I'd recommend a 180 grain bonded or partition in either the 308 or the 300 and in the case of the 300 I have had a lot better luck with 200 and 220 grain bullets. My experience with the 300 vs the 30-06 is that there is no difference at all in how well they kill, but the 300 gives you about 100 yards of extra range for an given point blank trajectory. Any 300 magnum works better not with more speed(the current fad) but with more bullet!
A 300 with a 220 grain bullet shoots plenty flat enough for elk out to 400 yards and a 308 with a 180 is just fine too, but you need to learn your hold overs at 250 and farther. Not as big a dea,l as some magazine articles will have you believe.
#1 elk vitals are a BIG target.
#2 If you know how to hunt, it's very likely you will get a shot at under 200 yards so hold overs are not even in the equation at all on an elk at that range.