Forget it; it works again. Don't ask me why.
When I posted this, I was at firmware 5.3.5. I've re-upgraded to 5.3.6, and still no go. While searching, I found a post by Kovid Goyal saying:
"You need at least one book with working page numbers from Amazon."
The only thing I did after that was re-download the English dictionary, and the Kindle User Guide from the cloud, and opened them. They had no page numbers... but then I rebooted the Kindle, re-transfered some of my books, and bingo: page numbers are back.
Could it be that opening an original / official AZW book "activates" the page count function of the Kindle, even if the book itself has no visible page numbers? Weird.