|Did you look in the Control Panel for Avira to see if there was any enable/disable setting for the tray icon?|
Sorry, but although I can see what you are driving at I don't think trying to restore the icon in TaskBar Properties will achieve the objective. Even if you somehow got the icon moved to Current Items there is no guarantee it would put it in the notification area (near clock) and it becoming the proper working facility that you need. It is not like an ordinary shortcut icon.
See if this helps (posts 9 & 10):
If not, there are lots of hits about "missing Avira taskbar icon" in Google (I only read until I got to the first one that "appeared" to work). If that doesn't help I think you would be better off joining an Avira forum because this seems to be a known Avira issue and someone there should be better able to help you.
Always pop back and let us know the outcome - thanks