While I have not mountain hunted as much as others, I have taken a few mountain animals over the years; sheep, mtn goats, caribou, wolf, and a grizzly.
To date I have used rifles chambered in the following cartridges: 6.5x55, 6.5 Creedmoor, 270 Win, 270 WSM, 7mm Rem Mag, 7MM STW, 300 WSM, 338 Federal, 338 Win Mag, 358 Win, and 375 Win.
One some, it was just the rifle I had with me at the time as I was backing up others on their hunts, and had my backup rifle (358), or coming back from work with my camp rifle (375).
These rifles have typically weighed from about 7 1/2 lbs to just a hair over 9 lbs scoped with full magazines. My preferred mountain rifles weigh from just under 8 lbs to 8 1/2 lbs scoped with full magazines (270 WSM, 7MM STW and 7mm PRC).
I like the flatter shooting sevens for their BC and retained velocity and energy for potentially longer shots, but to date, my longest shot to date was on my bighorn at 300 yards, while my first mountain goat was taken at just 15 yards.
The point is, any cartridge in a 6.5 caliber to the 375 caliber, in a rifle you are comfortable with and shoot well, will work on mountain game...the "best" rifle will be the one that fits you, and provides the performance that you are comfortable with for your intended quarry at the distances you feel confident that you can place the shot accurately in the vitals from a wide variety of field positions that you may need to use in the mountains.
While lightweight rifles are nicer for packing up steep mountains, they are more difficult to shoot when using field positions when out of breath, with your heart pounding after that final stalk in steep terrain to get into a shooting position.
Old Jack had it right when he recommended a rifle that weighed in at 7 1/2 lbs for mountain hunting. Light enough to carry, and heavy enough to hold steady when needed to make an accurate shot quickly with your adrenaline flowing.
I am also reminded of something a friend once said...drop 10 lbs of body weight and it makes it easier to carry an extra 10 ounces up a mountain! There's a hint of truth in that statement.