The gulf stream might originate in the gulf but the bergs originate in Greenland (north) and flow SOUTH in the southerly flow of the Gulf Stream. The Gulf Stream is a circuit, not a one-way flow. The bergs do not calve and then flow North to the Arctic ocean. They flow on the southerly circuit and melt as they float south into warmer waters.
The point you miss repeatedly is that they never ever flow toward the gulf.