So, in areas where brinicles do not form due to currents or depth, are there still descending plumes that carry the cold brine down from the surface levels? Is this what is meant by "brine rejection"?
Yes.
Brine rejection is the rejection of brine from the ice as it forms, because ice cannot contain salt. The first year ice contains a remnant of brine as brine pockets. But most of the brine is rejected, in models this rejected very salty, cold, dense, brine falls through the ocean column and plays a role in ventilating the ocean column, i.e. it falls through the layers of the ocean mixing as it goes.
The only reference to this process I've seen in observational studies is from the late 1990s.
Rudels et al, 1996, Formation and evolution of the surface mixed layer and halocline of the Arctic Ocean. It referred to brine rejection causing mixing at the shelf edge, the shelf edge being the edge of the continental shelf. The summer ice edge at the time was much closer to the Siberia at the time, which means less autumn/winter ice growth was occurring at the time. So brine rejection was much more limited in volume and was occurring nearer to the coast. In recent years there's been excessive open water away from the coast, even away from the continental shelf. I'm not sure what effect this is having.
Anyone got more up to date literature?