There is at least one e-mail provider for which I would love to archive all my e-mails into a package, save it to my computer, and then close the account--Yahoo. I want to be done with Yahoo e-mail after its displays of incompetence in providing security. (I just discovered Protonmail, but that's another issue.)

But I have never seen such a thing: software allowing me to collect and save my e-mails independently of the website. As far as I know, if it doesn't have an option to save my mail in a ZIP file or some such format, I can't have them; and as far as I know, Yahoo doesn't offer it. (I'm not aware Google does either.) Does anyone know a method? Other than the laborious effort of manually copying and pasting every e-mail?

(I realize that if I saved my e-mails and then "deleted" them from the website, they wouldn't actually disappear from its servers, at least not in any way I can verify. They may have my mail in perpetuity. But there's nothing I can do about it.)

Set up Thunderbird to POP3 the mail to your computer, then archive the TB profile.
That should also remove the mail from their server, but you can check via the web interface to confirm.

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Your mail client should simply receive the mails, remove them from the host server, and "archive" them any way that it sees fit.

If, for (say) corporate compliance reasons you need an assured archive, the easiest way to do this is to put a "mail proxy" endpoint within your organization which will make its own verifiable archive of everything that passes through – and to ensure that clients must use it. If you also arrange for that server to digitally sign outgoing traffic using a key that only it has access to, you establish a line-of-defense against external forgery of messages: "the message purports to come from us, Your Honor, but notice that it is not signed." You can't keep forgeries from being created, given the way that SMTP/POP is architected, but you can establish credibility.

If you regularly exchange e-mails with other domains who also have a known-to-you digital signature, a mail-proxy server can be set up to check that signature and to quarantine any incoming messages that do not bear it. Proxies can also be used to encipher and to decipher those communications, without placing that burden upon individual internal clients.

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I only just got around to doing this. I am now accessing my Yahoo e-mail in Thunderbird. What's the TB profile?

Never mind, I solved the issue without knowing what a TB profile is. All I did is move all all the "archived" e-mails back to the inbox in yahoo, and then download them in Thunderbird. Fortunately, I had no saved folders, because I wouldn't know how to download them.

It's the mozilla directory that contains the emails you've downloaded. By default it's in
~/.thunderbird/ and looks like 9x4acdg.default
that directory contains all the settings and emails you've downloaded.

Edit: You should be good to go, then.

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