How is Tbird configured to handle archived messages for the Gmail account? In other words, what is the current setting in Account Settings > Copies & Folders > Message Archives?

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You're using Local (offline) folder(s) as targets for an IMAP account. That's not going to work, especially with Gmail. Gmail doesn't even use true folders to organise mail. It uses "Labels", which is what you see as a folder in Thunderbird. Gmail uses the All Mail IMAP folder to store ALL your messages, and is also the correct Archive folder to use for archiving. Gmail correctly handles "moving" the messages from its own Inbox "folder" to its own All Mail folder.

OK Yes that's what I am doing and yes that's not going to work. Gmails archive tool does not separate out the archived files so I would like to use Thunderbird to do it. So rather than asking "does archive work on Tbird synced with gamil" I should have asked "is there a method to use the Tunderbird archive tool?" Because If I do used gmail as you suggest:

I would select messages to archive in gmail. As they are already in both gmail and Tbirds "all mail" folders; all that does is remove their gmail labels and remove them from the Tbird named gmail folders.

So now they are still in Tbird "All mail" folder mixed up with messages I have not archived. At this point in Tbird it is nigh on impossible to select them as a group and thus impossible to use Tbirds archive tool.

You have things mixed up. You said that archived messages did not get removed from the Inbox, but they are indeed removed from the Inbox. Where they don't get removed from when archived, is the All Mail (IMAP) folder. It seems you meant they were NOT removed from the All Mail folder. Remember, the All Mail folder is NOT the same as Inbox, so if you say "Inbox" while referring to the "All Mail" folder, we are bound to hit a communication obstacle.

Now, to answer your question, yes, you can use Thunderbird's Archive Options with Gmail, but the target must be a local folder. While in Inbox, archive a message(s) and it gets archived in the target local folder and gets removed from the inbox as expected, but the All Mail folder will still contain the message(s). If your intention is to not leave a copy of that message in your Gmail mailbox and keep the only copy offline/locally, then you must move the message from the All Mail folder to trash then empty the trash AFTER the archived messages have been copied to their respective local archive folders. The Gmail trash folder is automatically emptied periodically anyway.

While keeping the archiving options exactly as they were (in your screenshots), select a bunch of messages in the Inbox folder that you want to archive and hit the archive option. Confirm that the messages get removed from the Inbox and are filed away in their respective separate local archiving folders. Check in All Mail folder and confirm that they are still there, then switch back to Inbox and confirm that they are NOT still there. This is how it should work with Gmail.