This is a very interesting attempt at a workaround.
First, when I disabled animations, the usual "per icon fix" stopped working. Clicking on the invisible icon to restore the window and then minimize it no longer brings back the icon (so it only refreshed as a result of an animation).
Second, the whole system seems much snappier without any pesky (if cool-looking) animations happening. I might keep it this way and see how that works. Thanks, Craig!