My understanding is that disks use the inertia of their spinning platters to power themselves long enough to flush their cache in the event of sudden power loss.
They use this to finish writing the sector they were writing so they can guarantee they always completely write the sector or not at all.
As far as I know this mechanism is not used for write caches.
Yeah, I just had a look and it seems BBUs don't actually power the disk. They just keep the volatile cache alive until the disk is powered to flush them.