I should note that on this line:

error_page 503 /maintenance.html;

I have tried a variety of paths.
/maintenance.html
$document_root/maintenance.html
/some/other/path/maintenance.html

etc.

I don't understand how Nginx can see the stop.txt that I use to put it in maint mode, but it can't see (or ignores) $document_root/maintenance.html (when I had that set as the error_page)