It seems Dell Tech Support may have been blowing smoke. The REAL problem turns out to be Western Digital's WD Drive Utilities. I had installed that application and its associated services some time ago, thinking it was pretty benign. Uninstalling this app caused the problem to go away. Why an app intended to help manage WD drives finds it necessary to hammer a TerraMaster DAS with Seagate drives is beyond my comprehension.
Thanks, Igor, for the pointer to ProcMon. Though it took me some time to understand what I was seeing, ProcMon provided the essential bit of evidence pointing to WD Drive Utilities.