A bit of online research and some component testing narrowed my issue down to either a faulty ice maker or a bad supply valve. Appliance Parts Pros had both of these, at reasonable prices, so I purchased and replaced both items.
I recommend watching online videos to get a sense of what to do, and how to go about doing it.
The repairs were relatively easy, other than this being a bottom freezer model hence the ice maker was difficult to access: I removed the freezer door (four screws) to get closer, but never did figure out the slide-out shelf/basket component removal.
I wish I had a magnetized nut driver for all the times I dropped the ice maker screws! At least the water valve was an easy swap.