When diagnosing the problem, I discovered that the ice door was not properly sealing and that the solenoid had appeared to rust/corrode forcing the door open.
I replaced most of the ice dispenser internals (solenoid, crank, door, and spring) following the video.
The door will now open when dispensing ice (which it does!), and will oddly stay open for a few seconds once I am finished, but does seal properly and the chute/container no longer frosts over.