Failing that, just delete the install files and the configuration stored in your Documents\My Games\FINAL FANTASY XIV - A Realm Reborn

The game doesn't create or use any registry settings, except the ones that Windows uses to know where to uninstall the game (which clearly you don't have if you don't see it in the list!) That means the game probably wasn't actually installed on your PC (you can just run the game files without ever needing to install it, very handy if copying to/from another PC or if you want to reinstall your actual Windows installation)