I use a sling for every hunting rifle. Realistically, it never gets caught in thick brush and makes going through it easier because my hands are free to move vegetation out of the way while I stalk.. I sometimes put the rifle on my shoulder when crouching or stooping down to look at tracks or a blood trail, but immediately take it off my shoulder when I stand back up. Slings just make a ton of sense and have honestly never prevented me from hunting in any terrain (in fact they make it easier). When in the final phase of a stalk or while tracking blood, I carry the rifle in my hands (port arms style).
I cringe when I watch the hunting shows on TV and everyone decides that it is time to look "Cool" by carrying in such an unsafe manner when they go to Africa. The same guys have the same rifles slung on their shoulders when they hunt North America and New Zealand, but for some reason take their slings off and go full-retard when they get to Africa. Even Boddington seems to believe that the rifle he has slung while hunting pigs in Central California, needs to be carried "Africa Style" while on a ranch in RSA. This leads me to believe that most American hunters do it for nothing more than looks.
As for speed; if you are practiced, mounting the rifle from a sling is JUST AS fast as mounting the rifle from over-the-shoulder carry. So that argument just doesn't hold water for me.
As to PHs in Africa; I watched one episode where Ivan Carter probably muzzled everybody 10 times in a half hour show. He stood there talking into the camera with the muzzle of his loaded 600 Nitro Express pointed right at the cameraman's heart! American guides would come unglued if a client was so blatantly careless with a firearm.