For whatever it is worth, we are experiencing exactly the same problem. We have had no issue with any audio file for years. Suddenly, the player is missing from every post, replaced by the same thing that the OP has found. Even the links to "play in new window" do not work. They simply bring you back to the post page.
Changing the player to the HTML 5 option works, but that option does not allow the listener to pause, change volume, etc. Once the file starts the only way to stop it is to close the page. So, that is not a valid option. (Opening in a new window works, but most site visitors aren't going to do that. They are simply going to press the "play" arrow.
I do not believe that this is a theme issue, since our theme has not been updated recently.
Thanks for your help.