I did further research and tried to set up Archiving in Local Folders, per this article: https://support.mozilla.org/en-US/kb/archived-messages Even though the article claims archiving can help manage email server quote, I am observing the same results with gmail as by "Move To" - meaning that message moves from the on-line folder in Thunderbird to a designated archive folder and the label is removed for the message in gmail on-line, but... the message still persists in All Mail, thus still filling up the storage. Any help is appreciated.

To close the loop here, alas, I am giving up on Thurnderbird to use for Gmail. In general I think Thunderbird is one of the better email clients for Gmail when used with a new account or for current emails only. However, it is not suited for the off-line Gmail back up / archive use case. Besides two features above (Move To: and Archive) that don't work with Gmail, I also tried a more laborsome multi-step process of first Copy To: (local folders), Delete from on-line folders within Thunderbird UI, and then empty Trash. This was only partially successful until two actions. First I inadvertently rushed and hit Delete before doing Copy on a bunch of emails and had to fish for them in the Trash and put them back. Secondly after a small number of messages Thunderbird refused to copy more emails from on-line Sent folder to the local folder equivalent. There was no error. It simply wouldn't create selected messages in local folder. I realized it too late after deleting a bunch of Sent mail thinking it was already copied. I am not sure why this happened - maybe there's something special about Sent emails that makes them different from received emails and screws up Thunderbird by copying them into a local folder. Or maybe it is the sheer size of my bloated on-line gmail that Thunderbird 140.2 on Ubuntu is not calibrated to retrieve efficiently. Either way I ended up with a messed up on-line gmail after trial and error with the local Thunderbird client, including duplicated messages from trying to undo relocation of messages that were moved from folders but not deleted and then messages deleted while not actually copied locally, and see no point of trying to continue with Thunderbird. I have previously downloaded my Google data, including gmail, using Google Takeout, so in theory I have my gmail backed up before the Thunderbird experiment. My conclusion is that I will be deleting Thunderbird locally and just keeping my Google Takeout copy. I will then be deleting just enough of the oldest messages in live on-line Gmail directly from All Mail folder to make enough room to keep the account workable. Then at some point in the future I will do another Takeout and this is how I will have my gmail "sort of" backed up on on-going basis. I have not achieved my goal to be able to bring organized gmails by folder/label for convenient browsing off-line. So I am just left with the "blob" of Takeout data that I can theoretically use, if I really really really need to find something in the old messages. This is not ideal, but I haven't found any better way that is reliable and not very time consuming, alas.