Just following up on my fridge situation.
When the Whirlpool guy came over, the first thing he asked was what kind of water supply was I using. I was using an existing saddle valve tapped into my cold water pipe under the sink. He said that was the problem, as those needle valves only have a very small opening for water to flow through and eventually they get clogged with mineral deposits.
I replaced the valve the following weekend with a "non-piercing" (drill) style saddle valve and that fixed it!
This type of valve has a much larger opening relative to the tiny needle and we're getting great water flow and fully formed ice cubes now.
The Whirlpool guy was very cool and told me since there was nothing wrong with the fridge itself, technically they could charge me for a non-warranty service visit. He said he wouldn't even write anything up at all so I wouldn't get a bill.
Just as an example of how low the flow had gotten, I keep a 1-liter water bottle in the fridge and refill it when I drink it. With the old valve, it was taking close to 3 minutes to fill! With the new valve, I can fill the entire bottle in about 25 seconds (yeah, I timed it )