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Sorry to disagree but I do understand the physics involved very well.
The physics of anchoring a boat and a diver are identical, boats don't sink at anchor if it is done properly and neither would a diver.
The stronger the current is the more rope you need or the more lift you need.
Using your argument all buoys would sink if there was any current at all. As there is nearly always some current all buoys would be no use as they would spend there time at the bottom of the sea.
Take the example of very shallow water say 2 metres and 100 metres of rope.
The pull of the rope is almost parallel to the force generated on the diver by the current. So there is no downward component, so you would net sink.
Now take an example of the buoys you mentioned earlier where the rope is almost the same as the depth. in this case the robe behaves in a simillar manner to the punch bag, as the buoy gets pushed it immediatly starts to sink.
As I stated earlier the forces involved are
The vertical force = horizontal force divided by the tangent(90 - angle)
If you still don't understand then I sugest you look at
http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/u3l3b.html