SOLVED.
I just recalled that this happened once before, as posted here and here.
The solution then, as now, is to re-select the “Kee home group” in KeePass (right-click on a folder in the KeePass folder hierarchy > Set as Kee home group). Doing this restores the Kee extension functionality in both Firefox and Chrome, with immediate effect.
For some reason, the Kee home group sometimes seems to be forgotten, and without a home group, no matching entries are found. Perhaps it’s a bug that the home group is forgotten, or perhaps it’s unintentional user error, but whatever it is, the impact is sudden and annoying but the fix is quick.