Some very interesting answers so far,
Opposite Pole, but they puzzle me a bit because they are not what one would have expected.
I grew up around Chamois hunters in the Alps, then hunting Chamois, Mouflon and Bouquetin (Ibex) myself for close to 40 years, then hunting more recently mountain game in Africa (Vaal Rhebok and Mountain Reedbuck), and my thoughts are a bit different.
1- If you actually mean mountain game (Chamois, Mountain Goat, Sheep, Mouflon, Tahr, Ibex, Vaal Rhebok, Mountain Reedbuck, etc.) you can certainly use a 7 mm or a 7.62 mm (.30 cal) or an 8 mm, or an even bigger caliber if you want, but the grand classic mountain calibers (at least in Europe) have been 5.6×57 RWS, 6.5x54 MS, 6.5x57 Mauser, 6.5x68, etc. and the American most notable contributions have been .243 Win, 6 mm Rem, and .270 Win (curiously, the 25-06, a superb mountain caliber, did not really make it in Europe).
Do not read me wrong, I love the 7x65 R, and I actually own a lovely Zanardini kipplauf (single shot break open rifle) in this caliber, with a Zeiss Diavari Z T* 2,5-10 x 52 scope mounted in Suhl detachable claw mounts, with full length rib and iron sights, and it is a joy to carry and to shoot, but it does not really shoot very flat compared to the .24 (6 mm), .25 (6.35 mm), .26 (6.5 mm) or .27 calibers (6.85 mm) more modern calibers.
My Zanardini kipplauf in 7x65R with Zeiss Diavari Z T* 2,5-10 x 52 scope mounted in Suhl detachable claw mounts
2- Very few hunters I frequented in the Alps shot anything heavier than 130 gr. That was typically in the .270 Win. Old hands who used their pre-war 6.5x54 Mannlicher Schoenauer 1903 did shoot heavier bullets, but this was just because it was all that was available, and the typical RWS or Hirtenberger 6.5 mm loads used 105, 120, or 125 gr bullets. And these were traditional cup & core bullets... Those we used the .243 Win - or the better 6 mm Rem (the 7x57 Mauser necked down to 6 mm, therefore essentially a 6x57) - typically shot 100 gr bullets, and while it lasted, the .270 Win load with 100 gr was considered one of the deadliest loads.
My Mannlicher Schoenauer M72 in 6 mm Rem (6x57) with Swarovski Habicht 2,2-9 x 42 scope mounted in Mannlicher detachable swing mount. The quintessential European mountain rifle... Notice the enlarged trigger guard to allow thick winter gloves (with a slit on the front face to allow good trigger feel).
Again, nothing wrong with the 7x65R but objectively, there are better mountain caliber, when it comes to flat trajectories, and mountain game typically does not require big calibers and heavy bullets, although of course they will work
3- In terms of scope magnifaction, I have yet to understand how can one need more than 10x on a big game rifle, but admittedly the fashion is toward ever bigger and more powerful, which is fine and dandy until you end up desperatly searching for your animal in a mega magnification scope cranked all the way up and offering, by force, a very narrow field of view...
In conclusion,
Opposite Pole, I would think that in term of "modern" flat shooting calibers, one would be hard pressed to get a better mountain rifle caliber than, take your pick: 25-06, .257 Wby, .26 Nosler, 6.5x68 (for our European friends), etc. The 6.5x55 will always be in the run, but its trajectory suffers a bit at longer range. Some will suggest the currently fashionable 6,5 Creedmoor, although its reputation is about clicking scope adjustments and punching paper at set distances - which is a little different from mountain hunting - but for my money it does not shoot flat enough.
I am personally in love with the .257 Wby with which I just took 17 animals in Africa last month, from Impala to Roan, from 100 to 400 yards, with 100 gr TTSX, all one shot kills (see
https://www.africahunting.com/threa...ill-safaris-even-better-than-last-year.52376/).
See Ron Spomer's 2018 piece on what could apparently serve you well: single shot Merkel kipplauf in .257 Wby. I think that AH does not allow posting links, but you can google it under "SPEED DEMON 257 WEATHERBY MAGNUM." My experience, like the experience of many, is that the little .257 WBy / 100 gr kills way out of proportion with its diminutive size, especially with modern mono-metal slugs such as the TTSX, the ETip, etc. and it is almost recoil-less which is a blessing in a light mountain rifle...
PS: PM me if interested, my Zanardini kipplauf may be for sale to the right buyer. I have a 2020 safari to fund