Few different views, in this thread, depends on what type of conditions, you are hunting in, OP said mountain hunting, that to me means, high, steep, rocky crags, low stunted bush and scattered scrub, with the odd patch of forest in places, and alpine grasslands, and scree.
In such conditions I like a properly set up light weight sling,
its a shooting aid, frees up both hands for extra safety in steep or unsafe conditions, and the option of letting the shoulder, take the weight at the end of a long day,
I prefer a 1", straight light leather sling, long enough, to do two wraps around the forearm for prone shooting of the elbow's or sitting and resting of the knee shot.
The wide padded sling, so common in most stores are bloody useless, as a shooting aid, and I find slip off the shoulder much easier.
A good sling on my rifle, likely saved my life, many years ago, on a bluff, high in the southern alps, after getting in to trouble, chasing a chamois around a bluff.