It's pretty clear to me that, to get the full cleaning effect intended
one must put the tablet in the *uncovered* cup, or into the bottom of
the dishwasher, where it will fall as soon as one shuts the door.

Maybe so; but that's not what the instructions say to do.

I pick up a package of those tablets to keep around: they don't take much
space and they are insurance when I run out of the cheap powder I usually
use. (God bless Dollar General).

For my dishes and pans I get the best results by using the extra long
"Pre-Wash" cycle on my machine. It's the equivalent of soaking the dishes
in detergent water for 20 or more minutes. With the tablets there would
not be any detergent in that first, long cycle.