I have downloaded and installed all the plugins from the list above. But I could not reproduce the problem. But most of the plugins have a lot of options or require a WP.com account (Jetpack).
Because it would be difficult to exchange all your settings in order to rebuild the situation in your blog, I would like to ask you to test the plugins yourself.
Activate the stats feature again for this test. Deactivate one plugin, reload the page and check whether the download link works or not. If it works then you know that the deactivate plugin was responsible.
BTW: Tumblerize produces an error on activation and WP does not activate it.
One other thing, can you embed the podpress from one site on another?
You can specify a full URL in the Location field in the podPress box below the post editor. This URL may point to a file on a different server or website.