This seems to have happened since the WordPress 4.0 upgrade (I think). When I use the "Add Media" button for an audio file, I can see the player in the WP backend, and it functions. But it doesn't show up on the page at all. I noticed this with a couple plugins that call the built-in media player, but in testing, it doesn't work even with a simple insert, as noted.
I do have another site (on different server, mind you) running WP 3.9.2 where the media player is working fine.