set the AMD HDMI Audio as the default playback device in the Windows playback properties
I have this exact problem and your solution isn't an option at all. The AMD HDMI Audio playback device is greyed out and so is the option to set it as default. No matter what I do (enabling/disabling, restarting, unplugging/replugging, switching outputs) nothing works. If I go into CCC it says that the display device I have connected by HDMI does not support audio, but it worked fine for my old 5670. I updated the drivers to their latest version, downloaded the Realtek ATI HDMI Audio separate driver from their website, newest Catalyst, and still nothing.
I eventually fixed this problem by simply updating the drivers. I first uninstalled all pre-existing AMD drivers, then downloaded the newest off their site, chose custom installation, and selected everything that wasn't installed. It had the option for AMD High Definition Audio or something like that. Maybe "AMD HDMI Audio" was what it was actually called. Well, when I did that, installed, rebooted and everything worked perfectly.
I just fixed the same issue on my system. It ended up requiring me to uninstall my realtek drivers before the ati installer would reinstall the audio drivers. Installing the realtek drivers after the catalyst drivers appears to be what caused my issue. So no more realtek drivers and it works great.