You’re right, cold water doesn’t rise. But when it sinks, it is replaced at the surface with warmer water, which thus creates a circulation.
I see this article stating a causal relationship: colder temperatures in the deep sea is causing oceanic masses to circulate. I just don't see how.
Water is densest at 4 deg C; above that temperature, it is less dense, and below that temperature - down to 0 deg C (where it will form ice) it is less dense.
This is why ice forms on the tops of lakes; the coldest water is at the surface, and does not sink to the bottom. Otherwise your lakes would ice up from the bottom.
So it depends upon the surface temperature. In the Antarctic, most of the surface water is at 0-2 deg C, and is less dense than water that is at 3-5 deg C, meaning the warmer water can be underneath the surface.