The only way to continue working is to leave the pop-up open and click on another window, giving it the focus. I set the Sonic WALL packet monitoring for all packets.
During all this time, Outlook continues to indicate that it is connected to the Exchange server and to download messages, but it will not automatically download messages from POP3 servers that are also configured. I started capturing, clicked OK in the pop-up window to send the stored password several times, and then turned off capturing.
Then I turned on a laptop with Outlook 2016 installed which, for a reason I don’t fully understand, had not updated its internal clock to the server’s time. And then it showed up on one of the formerly fixed Outlook 2016 clients.
I rebooted the machine until it picked up the correct time, set it ot “work offline” mode, and dis the dame thing to the other Outlook 2016 client that had caught the disease. So I think the root cause was the wrong server clock, which caused the client’s clocks to be wrong, which caused Exchange to reject part of the automatic client login (but, strangely enough, not the whole thing).
I don’t know why toggling “work offline” fixes thing, but it appears to.