It is a dynamic system with many variables, yes those set up the broad conditions but the geography and role of wind as well as crucially less saline water in the north atlantic creates a very nuanced and sensitive conveyor belt, far more than you might think when people describe it as a current.

Warm salty water flows north in the Gulf Stream, and then sinks when it cools in the North Atlantic (because saltier water is denser). Once it sinks, it flows back south along the bottom of the ocean.

Cold fresh water entering the North Atlantic due to ice melt from Greenland disrupts that process. The faster Greenland's ice cap melts, the slower the current gets.

If it stops entirely, heat stops getting transported by the Gulf Stream and northwestern Europe (especially Britain) starts looking more like Alaska.

It's the ice melting. When the Atlantic current gets to the Arctic it's extremely salty and sinks to the bottom. That's one of the primary engines for that current.

When the ice melts it dilutes the salt and so it no longer has the salinity to sink. That would make it stop!