DIAGNOSIS You're right—the main water reservoir is not designed to be removed. The cleaning process is a flush-and-drain system, not a disassembly.
FIX The fastest method: Unplug the unit. Use the included scoop to remove any standing water. Mix 2 cups of warm water with 1 cup of white vinegar. Pour this into the reservoir. Use a soft cloth or sponge on a stick to wipe the inside walls. Then, plug the unit in, turn it on, and let it run a full ice-making cycle with the vinegar solution. Once it produces a batch, discard that ice. Finally, run two more cycles with fresh water only to rinse.
IF THAT DIDN'T WORK For stubborn pink biofilm or scale, use a dedicated ice machine cleaner (like Affresh or similar) instead of vinegar. Follow the product's dwell time instructions precisely before running the cycle.
HEADS UP Never submerge the base unit in water. Do not use abrasive cleaners or brushes on the silicone ice tray—it will scratch and cause ice to stick permanently.