What is needed is its complete model number as was noted in the first forum message
// It's a Whirlpool ET18GK. The label says "accepts ICEMAGIC icemaker," which I'm assuming is the installed unit - I'm not the original owner so I don't know for sure.
** the unit makes more or less continuous click-like sounds **
From where *exactly*?
// From inside the motor/gearbox casing, i.e. directly behind the white cover in the front. It's not really a clicking sound though. Very hard to describe. It sounds like "gears are meshing trying to make ice."

It's mostly a continuous noise stopping periodically, for up to a minute or so, and then starts chirping away again. It continues to make noise whether the wire is up or down. I expected it would stop altogether with the wire up (which I left up overnight). Maybe the icemaker thinks it has a batch to expel and, otherwise, would have shut off?

BTW, this noise is NORMAL. The icemaker would periodically make exactly these sounds shortly before ejecting a load of ice. The problem is that it's mostly continuous now, and no ice is produced.
** I tried filling the mold with water (no change). **
Did it freeze???
Yes. The freezer is definitely cold enough.
Thanks for your help Dan O. !!
lloyd