Here's a more complete explanation and update on this issue.
Note that this issue doesn't happen with almost all Windows apps I use. Around 85% of my apps with tray icons restore their icon normally.
Every now and then, I have a need to restart Windows Explorer, whether by me manually restarting it in Task Manager, or there's an app installation that automatically does the restart.
Upon restarting Windows Explorer, Input Director's tray icon disappears and doesn't come back. Re-launching Input Director (e.g. double-clicking a desktop shortcut) sometimes brings it back, but also sometimes generates an error about not being able to run a duplicate process. In the latter case, I have to go into the Task Manager, end Input Director's task, then re-launch Input Director to get the tray icon back.
Current Windows version: Windows 10 Home Version 21H1 (Release Preview Channel), Build 19043.1200, Windows Feature Experience Pack 120.2212.3920.0.
I realize this is a minor issue, but it does take extra work to fully restore my system after app installations because of it.