That is a component of Windows Media Player, not FFXIV... so reinstalling the game won't necessarily restore it. It's <routinely> installed as part of the core Windows environment, so the game likely won't install it... probably doesn't even check for it.

The file may actually still be there, but the registration is somehow monkeyed up. Running an update to either WMP, MSIE, or even just Windows Update may restore the file or keys.