Since I learned to shoot before I learned to hunt, Rhodesian carry was favored. It was simply called African carry back then, but now African carry means that silly over-the-shoulder stuff. Also called Marine carry, but rarely in serious shooting circles. The advantage is complete control over the gun, as your left hand is controlling the forend all the time, while at the same time being very fast into action, as described above by others. Very low profile way to carry. Rhodesian carry is very applicable to guys carrying sidearms (obviously not usually visiting hunters these days).
American carry is muzzle up on the strong side. Slow to get into action compared to Rhodesian, it is nonetheless needed to spell your other shoulder from time to time. Right hand remains hooked into the sling to keep control. When you mount it you press forward with the right hand and the left hand comes in and grasps the forend. Medium-high profile.
Capstick carry, usually called European carry, seems to work fine too, but I personally have never warmed up to it. Very high profile.
When carried correctly, the sling will not slide off in any of these positions, as you have control over it with a hand at all times. All of my guns that are capable of it, have a shooting sling attached. I don't have much use for the so-called hasty sling, but a Rhodesian sling, as made by Andy's, is a very fast way to loop up for longer shots. Proper schooling and practice is required in order to get the most out of this.
In thick brush, I will always have the gun in hand, and even out in the open, I often prefer it. I also prefer slim lightweight guns, and as much as I like slings, all of mine have QD swivels so I can ditch it if I want to.
If some PH or guide tried to tell me how to handle my weapons, I'm pretty sure my diplomacy would be tested. I'm sure there are some out there who can shoot, but at least from the videos I have watched, most don't seem to have very good gun handling, and with the popularity of shooting sticks, I don't suspect that real shooting ability will come back into vogue over there. I well know the value of shooting sticks (especially when someone else is carrying them for you, kind of like your kindergarten teacher used to do for you), but it always cracks me up to see a client getting on sticks, when there is a tree 1 foot to his side.
Go take a 270 rifle class from Randy Cain or similar. Your hunting experience will improve dramatically, as will your shooting ability.
Obviously this is a generalization, and not aimed at anyone here. Just part of my frustration that a nation of riflemen has fallen so far.