The best “out west” big game cartridge is of course the .308 Norma - my user name. So far, I’ve killed 2 mule deer with my Norma, each 1 shot kills. And both deer dropped in their tracks.
I’m being flippant of course. I love my .308 Norma Magnum, which I had built by Montana Rifle Company as a retirement gift to myself. But the fact of the matter is I killed my first mule deer with regular .308 Winchester. And that deer has been dead for well over a half century. In that half century, I’ve killed mule deer, elk and antelope with everything from a .30 Carbine to a .338 Winchester Magnum in terms of power, and everything from .223 to a .50 in terms of caliber. They all work on western (Idaho) big game. It’s just a matter of putting the bullet in the right place.
Of course the bullet has to be heavy enough, and be traveling fast enough to penetrate to something vital. That said; one of the largest deer I ever killed was the one I killed with a little .30 Carbine. Don’t get me wrong – I’m not advocating the use of a .30 Carbine for a deer rifle. Without getting into the whole scenario, I’ll just say that the deer I shot with a .30 Carbine was at the most, 10 yards from me, standing still, and I put a 110gr., .30 Carbine bullet right behind his ear. He wouldn’t have dropped any faster if I’d have shot him with my .308 Norma Magnum. But for all those who claim “bullet placement is everything” because that deer would have dropped just as fast if I’d have shot him with a .22LR (illegal for deer hunting in Idaho anyway) - being no further away than that deer was from me, I could have easily placed a BB from my Daisy Red Ryder in the same place (right behind his ear) and he would have just ran off.
To get back the OP’s question though, for the “best” all around cartridge for big game hunting “out west,” the good old 30-06 gets my vote. But my dad killed a lot of deer and elk with his beloved .308 Winchester. So I wouldn’t rule out a .308 Winchester either. And while I’m not a fan of the various 6.5’s, I’ll admit that with today’s powders and bullets, the .264 Win Mag might be another good choice. So might be the .270 Winchester or the .280 Remington. I think both the 7mm Weatherby Mag and the 7mm Remington Mag are good options too, with slight favor going to the Weatherby if you’re a handloader. Any one of the various .30 caliber magnums would be a great choice in my opinion – again, if you’re a handloader. Finally, for an “all around” cartridge for big game hunting, not just “out west” but anywhere in north America, there’s the .338 Win Mag. A .338 Win Mag will easily spit out a 225 grain bullet out at 2800fps+, just like a 30-06 will do to a 165 grain bullet. Furthermore, ballistics charts show that the trajectories of the two bullets are the same clear out to 500 yards. The trade-off is of course recoil, although I personally never thought my .338 Win Mag kicks all that hard.
Anyway, like I said, I love my .308 Norma Magnum, but the 30-06 gets my vote for the “best” all around cartridge for big game hunting out here in the west. The .308 Norma pretty much requires handloading and a custom rifle. Besides, I seldom load my .308 Norma to its full potential anyway. My handloads for it are more like hot, but still below maximum, 30-06 loads.