How can I setup Thunderbird such that the emails that are successfully filtered to the different accounts will be deleted from their "All Mail" folders?

You can't. Gmail literally stores all messages in the "All Mail" folder - hence the name. Learn and understand how Gmail labels work. https://support.mozilla.org/en-US/kb/thunderbird-and-gmail

You can unsubscribe from the "All Mail" folder, so that it is not shown in the Thunderbird folder pane.

My first suggestion is to consider whether you need the Allmail folder. That is a gmail concept, and it is the repository for all of your gmail messages. To make this concept work in an email client such as thunderbird, the messages are duplicated into Inbox, sent, etc. That begs the question of what purpose Allmail serves. If you remove Allmail, your incoming and outgoing mail will be faster and your hard drive use will drop noticeably. To answer your question, to get the message removed from Allmail, I would update the filter to COPY to the other folders and then DELETE from allmail.

to get the message removed from Allmail, I would update the filter to COPY to the other folders and then DELETE from allmail.

Not sure what you mean. Mail clients cannot remove messages from the All Mail folder, only Gmail can. It's not a matter of telling the mail client to delete instead of move or whatever, it simply can't no matter what.

The most you can do with a mail client to delete messages from Gmail is move them to the Gmail Bin/Trash folder, so that Gmail (again not the mail client) permanently deletes them from the server after 30 days.

AllMail is a folder. If you click a message in the Allmail folder, you can delete it.

LOL. Nope. Neither is All Mail a normal folder nor is Gmail a normal IMAP server… Mail clients CAN'T remove messages from All Mail, only Gmail can…

Isn't that semantics? Thunderbird is an agent. A user of thunderbird, over time, causes messages to be deleted. If I click to the allmail folder and click a message to delete, it goes to trash.

This is not a matter of semantics.

First off, what happens to a message as a result of you telling the mail client to delete it depends on your account settings. Gmail recommended settings for Thunderbird do even (incorrectly) tell users to not move deleted messages to Bin/Trash and just mark them as deleted instead. And they say the following for mail clients in general:

Don't change your IMAP settings to save deleted messages in the Trash. When you do this, any messages you delete from your email client will get permanently deleted from Gmail in 30 days.

So they tell users to configure the mail client so that deleting a message there does not work the same as deleting the message in webmail, and the reason is… that if the user does otherwise the message will get permanently deleted after 30 days, just like it would if it was deleted in webmail… This is absurd, but is what Gmail recommends, so many users may do that not really understanding why.

Second, we were talking about removing messages from All Mail, remember? Not from some other folder. So even if you do correctly set up the mail client to move deleted messages to the Trash, they will still remain in All Mail for 30 days, so the fact remains that you cannot accomplish what you wanted when you tried to delete the messages (i.e. remove them from All Mail) until 30 days later. And that's assuming you don't make the mistake of configuring the mail client so messages get permanently deleted from Trash before the 30 days that it takes for Gmail to do it, because if you do that what will probably happen (haven't checked, but is what I would expect knowing how Gmail works) is that the Trash label will be removed from the messages and hence never be deleted from the server…

Now, IF you configure the mail client properly (i.e. not following Gmail recommendations) AND don't subscribe to All Mail AND don't really care that you cannot make messages to be removed from the server until 30 days later AND make sure you always do copy+delete instead of just moving, THEN you may find that Gmail works as expected, but that's because you've adapted your expectations to the way Gmail works, not a matter of semantics…

I believe it is just semantics. I cick a message in my gmail account, click to delete it, it goes to trash, I empty trash, log onto gmail online, and the trash folder is empty and the message is not in allmail. I will agree that, depending on user setting, that gmail may keep it for awhile, but that's user control. Also, when deleting from thunderbird, thunderbird no longer 'sees' the message, and that satisfies the intent of the original poster. And I will leave this thread, as continuing it serves no one. Thanks for the discussion,

Interesting… It's not that this is a matter of semantics, it's that I was wrong about what happens when the mail client deletes a message from Gmail Bin/Trash. I thought doing that merely caused the Trash label to be removed from the message, but looks like the removal of that label in turn triggers the removal of the message from All Mail. I was unaware of that, doesn't seem to be documented anywhere.

So, contrary to what I said, mail clients do have a way to delete messages from Gmail, namely by moving the messages to Trash first (so the Trash label is added to them), then removing them from there (so the Trash label is removed, thus causing the messages to be removed from All Mail)…