Although I repaired the spout a while ago, those water bottles were never satisfactory and saw very little use. A recent cabinet cleanout showed the “stainless steel” has passed beyond its best-used-by date:
With no regard for whether the patient would survive the operation, I peeled off its rubber foot and applied the Lesser Hammer:
The “insulation” seems to be a rigid urethane-like foam disk few millimeters thick on the bottom of the interior flask, with good old air around the sides.
The bottles never worked very well and now we know why.