After reading multiple discussions and websites about the possibility of certain refrigerator reservoirs breaking down in chlorinated water causing tissue-like little flakes in the chilled water, I dished out the bucks and ordered a new coil-type reservoir and installed it.
Well, weeks later and after countless gallons having been run through, I see no difference. Flakes.
When I took out the old reservoir, I emptied it, filled it with water, shook it out, and inspected the water that came out. No flakes.
The flakes for me are not coming from the reservoir. I don't have or need a filter, so they're not coming from that either.
No solution yet, but at least I know it wasn't the reservoir.