Thunderbird does actually show a red dot on the system tray icon at computer startup to indicate unread emails. But you'd have to make a screenshot of the desktop and put it on an always-on-top HTA window, to reliably hide the white thunderbird window that appears at computer startup. However then you'd still have the issue sometimes where you need to click the taskbar icon twice before it actually opens. My workaround for that wasn't reliable and I couldn't come up with a better one