I know this is not what you are saying, but the presence of absence of an icon on the windows taskbar is not evidence either way of the software being installed, only the presence of absence of a taskbar icon being visible.

Why yours disappears I really do not know, have you searched Thunderbird in the windows task bar search box? Are you offered an app in the results or to search the web.

For example I have more "Thunderbirds" installed than appear on my task bar and it is not an accident, but search finds them.

Oh yes, brilliant, of course that could be an easier solution! Thank you. It doesn't resolve the icon disappearing from the bottom task bar (still baffling!) but is much simpler than installing Thunderbird again and again. Thanks :)