Update: I've just starting backing up the new machine - using the original Time Machine backup of the old machine. I wanted to make sure, this time, that "Encrypt backups" was turn on. Guess what? It's turned on.
So, now I'm confused. Was it turned on the entire time? I'm not really sure. When troubleshooting with Apple over the phone I did not have my external backup disk used by Time Machine plugged in. I determined no encryption was used without the backup disk plugged in. But, now I believe I probably should have had my HD plugged in to say whether or not encryption was used.
I think I had encryption on the entire time.
Yet, it's a total mystery why my passwords didn't get restored (as they should have been). In a roundabout way Apple strongly indicated that passwords weren't backed up because I didn't use iCloud. Chrome had zero issues with passwords restored. I get the impression Apple wants to force iCould on users - punishing those that don't use it by not restoring passwords. I'm pulling my hair out.