The Gulf Stream doesn't shut down. The AMOC shuts off or decreases in intensity. This is thought to have led to a number of interesting climate blips since coming out of the last ice-age. i.e. melt water from North American land sheets flows into North Atlantic, making it less dense, and less likely to sink. This leads to a lessening in the AMOC. The Gulf Stream would have still flowed but perhaps not as intense nor have been "pulled" as far northward. One such event was 8200 years ago [0].