Oops, sorry about my brainfart. Of course it does not shut down the onion service in case I only download the HTML download page (using wget), but not the payload itself. After downloading "/download", onionshare says "Closing automatically because download finished" and the hidden service is gone (but it does not seem to exit until I hit ^C). --stay-open works as documented.
However I still cannot get the "stop server automatically" checkbox in the GUI to work as expected.