When you get seemingly odd behavior like this, it's always a good idea to check the Network tab in Chrome Dev Tools (or Firebug in Firefox or whatever it's called in IE/Edge).
When you click the "Generate ZIP" button, the Sitecore client makes a POST request to
/sitecore/shell/Applications/Tools/Installer/Designer. Because nothing was logged to the Sitecore log on the server, I suspected that the response from the server would have some clues.
In this case, the response had a HTTP status code of
200 OK, which threw me off at first. However, inspecting the actual response showed the problem:
"value": "Cannot access path 'D:\\site\\web\\Data\\packages'. Please check PackagePath setting in the web.config file."
I'm not sure why Sitecore returns a
200 OK here instead of a
500 Server Error, but that JSON clearly shows the problem in my case. This dev's local environment was missing the data\packages folder. Creating a new packages folder solved the problem (I encouraged the dev to copy a fresh Sitecore installation on top of the existing folders to ensure no other important items were missing.)