Following the threads I had seen on the Garmin forum, I tried a few of the suggested "fixes" all the while waiting for a response to Customer Support. Finally got a response today. Once they were "on the line", things progressed rapidly, though still to no avail. The damn thing is still dead in the water.
Garmin is sending me a new charging clip, believing that to be the issue. I withhold judgement until it gets here and I can give it a whirl. The next step after that is to replace the unit, which I've had only for a couple of months.