On the memory trail...
The quintessential "mountain rifle" of the 1970's and 1980's in Western Europe was the Steyr Mannlicher with Stutzen stock. The two dominant calibers in France were .243 Win and .270 Win, while in Germany and Austria one would see just as commonly, and maybe more commonly, the 5.6x57, 6.5x55 and 7x64, and in Italy it would be a draw between American and German calibers. Eastern Europe, behind the Iron Curtain, was a mystery. The .25-06 was virtually unknown in Europe, at least to my knowledge, which is too bad as I reckon it would have been quite something in the mountains. The scope would almost universally be either a fixed 6x42 or a variable 1.5-6x42 Zeiss or Swarovski.
Mine was a Steyr Mannlicher Luxus Stutzen in .270 Win with Zeiss 1.5-6x42 on Suhl claw mounts. I still have it, and this is still my favorite mountain rifle for sentimental reasons.
Chamois hunting was generally a fall hunt, with winter snows starting to come down the mountains.
The holy grail was a grand old Class III Chamois
There were still a fair number of Mannlicher Schoenauer M72. This was a great rifle, but a lot less sophisticated than the Steyr Mannlicher that succeeded it. I had the opportunity to acquire one in 6 mm Rem. It was bullet proof and featured a large trigger guard to allow heavy winter snow gloves (with a small slit cut for the finger pad to contact the trigger). The 6 mm Rem - essentially a 6x57 - was almost as common as the .243 Win in those days in Europe, and generally recognized as slightly better.
And one could even, from time to time, bump into an old timer carrying the mythical Mannlicher Schoenauer 1903 in 6.5x54. I had the opportunity to acquire one, but alas, without its original scope. The 6.5x54 was a good caliber in mountain hunting, but its trajectory was arcing, and its range was limited compared to the more modern calibers.
Gun madness was upon me in those years, and I talked myself into a single shot Kipplauf, a delightful Italian Zanardini with 2.5x10 Zeiss on claw mounts. I took it in 7x65R because I wanted the possibility of heavier bullets for Red Stag and Sanglier (Wild Boar).
If anyone wants it, PM me...
Today...
I reckon that the best calibers for lighter mountain game (e.g. Chamois, Vaal Rhebok, Mouflon, etc.) are the American .257 Wby and German 6.5x68 S, as well as recently introduced similar offerings such as the 6.5x65, 26 Nosler and various PRCs, but I shall confess to not knowing them as I see no reason whatsoever to replace the .257 Wby 100 gr TTSX that has become my favorite mountain and light PG caliber.
Heavier mountain game (e.g. the large Asian and American sheep) require more power. I can see the point for 8 mm, .338, etc. but I am still not on board for 800 yard game sniping, so although I own a .340 Wby which is the ballistic twin of the more recent hot .338s, you will not see me on a mountain with it. It is my Moose gun. The various American .300s and the German 8x68S top the list of what I reckon to be the best calibers for heavier mountain game. My big sheep, and large PG caliber is the .300 Wby 165 gr TTSX.
As to the rifle, I think that I am likely to finish my life with the Blaser R8. Since I prefer an additional pound on my shoulder to a few stiches on my eyebrow, the ultra-light argument does not resonate with me, and anyone who has actually shot upward in near vertical terrain when the scope comes so close to the face will understand exactly what I mean, and anyone who has actually hunted difficult mountains has come to realize that a little rifle heft helps tremendously when the wind blows and/or one is out of breath...
Final thought...
Do not deprive yourself of a mountain hunt in Africa! Mountain Nyala in Ethiopia may be a bit pricey for most, but Vaal Rhebok and Klipspringer in the Eastern Cape are a delight...