I suspect there is truth in a number of these observations, such as the wood adding balance to a short rifle and also perhaps protection (or the appearance of it), to a small barrel, (small by the standards of the day). Keep in mind that the 6,5x53 MS and later 6,5x54 MS were very early smokeless rifles and, being much smaller calibers than many of the 10mm, 11mm and etc., black powder equivalents of the previous era, they may have looked a little flimsy, so a little extra wood may have conferred a stronger appearance to sceptical buyers.

However, I would have thought that, if hunting chamois in the Alps that a very real possibility was that of fingers sticking to frozen steel, so the wood acted as protection when the rifle was held near the muzzle end for any reason.

The extra wood does create problems with point of impact. For instance, European full length rifles tend to be slim but copies of them made elsewhere tend to be less so. So if the wood warps, it can influence the barrel. A partial (?) way of solving this problem was to make the full-stock extension a separate piece of wood that had a subtle join of some kind in the middle. Another consideration was the stirrup mount for the front of the sling. Some of these attached through to the barrel as well as the stock and this could create problems, or at least, so the advertisers offering alternatives suggested.

I suspect another reason for their popularity is simply that they looked European and this appealed to buyers. There are even 2 and 3 barreled guns for sale in Germany with full stocks ! The fact that Mauser made a copy of the Mannlicher Stutzen suggests that it had a faithful following.

Re the leopard blind; I was looking at a Sauer Model 90 in 9,3x62 with a large objective scope and an illuminated reticle. Being a Stutzen, it was quite a bit shorter in the barrel than my .375H&H of the same model. It seemed ideal to me. However, the extra muzzle flash might make it harder (or perhaps easier), for the PH to determine if a good hit was made. Short barrels also tend to be noisier, all else being equal.