As long as the winds blow and the Earth spins, we will have a Gulf Stream. What the paper is referring to is the three-dimensional circulation driven by salt and heat fluxes. Sinking cold water near places like Iceland and gradual upwelling of hot water in the tropics. Net result is transport of heat from tropics to poles and since it is water with a high heat capacity, a lot of heat! If this 3D flow is lessened, it means other things must adjust to compensate --- like midlatitude storms etc. The AMOC also "pulls" the Gulf Stream more northerly before it closes the loop and comes down the coast of western Europe. So UK, Ireland warmer than you'd expect given the latitude. Clear as mud?
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].
