Are you using a secure mail key in TB in place of the account password?

https://www.att.com/support/article/dsl-high-speed/KM1010523/

How can I tell?

If the entry in Saved Passwords is the account password, it's not a secure key. Create a secure key on the AT&T site and use that instead of the account password.

In either case, where are those entered (since there is no prompt)?

If there's already an entry in Saved Passwords, right-click, Edit. Or, delete it, restart TB, enter the key when prompted.

This for 'normal password' authentication. I think OAuth2 authentication might be supported for AT&T in TB 140, in which case enter the account password in the OAuth window when prompted.

"If there's already an entry in Saved Passwords,........"

That is the whole point. Please read first post, where it states: "Deleted passwords and cannot see where to create them again, as it never prompts me for new ones."

What is the authentication method on the incoming and outgoing? Confirm that before trying to figure out why you aren't prompted. I'm assuming there is no VPN or security software in your setup. Also, confirm your other server settings are as in the support article.

All confirmed. But please give reasoning for comments.

The reason? To eliminate the possibility the issue is caused by a simple factor such as an incorrect setting or an interfering external app. Is your current profile one that has been used in many earlier versions of TB? Sometimes the presence of an obsolete file can cause passwords to malfunction. Adding the account to a new profile is one way to test.

Okay, so sure, I already understood all that. However, the idea of the presence of an "obsolete file" is a little odd. The upgraded programming of each new version is supposed to take care of that, or the programmers will have a lot of problems. The fact is, I've never run across such a thing, and, in any case, it does not appear to have anything to do with my described situation. The passwords are not malfunctioning; rather it appears that the program is. It seems to not know that there is no password present, and so attempting to connect without recognizing that fact is an indication of a serious problem.

Oh, wait a minute, that clearly indicates a bug. I need to report that.

Thanks.