OK, after some cursing and swearing and so on and so forth, it turns out that some of my new WordPress themes did not include the WordPress.com Stats code in the footer.php:
<?php wp_footer(); ?>
The stats have started to return after I inserted that line of code, just like magic, duh. It seems that I learn something new every day 
Oh I’m so proud of you! You found missing code and you FIXED it! Now – will you admit that amidst the swearing (and so forth) it is actually kinda FUN and gratifying to fix it and make it work?
I wouldn’t say it was fun (yet), but it is indeed gratifying to be able to fix these buggers.
This was a pretty annoying gotcha. I’m glad that you found it and published a fix.