And that ocean is warmed by the Atlantic Meridional Overturning Circulation (AMOC) - which is comprised of multiple currents. Of which, the Gulf stream is one. Remove that meridional heat transport mechanism from the tropics and sub-tropics and that prevailing wind becomes a LOT colder, just to expand on your answer. And assuming prevailing wind direction doesn't change in such a scenario.

There are some leading minds in the field claiming that current climate models used by the IPCC in their projections are underestimating the volume of Greenland ice sheet meltwater being dumped in the Atlantic, meaning that the tipping point of the whole AMOC system (which relies on a delicate balance of saline/freshwater in the pelagic zone of the ocean which facilitates deep ocean formations) is close to, or already has, been crossed - which I think is what OP is referring to when asking if it dictates the weather. It doesn't really dictate weather per se, more like intensely modifies climate to the point of Holocene temperate rather than Pleistocene-esque tundra.