It turns out that I didn't need to wipe my browser installation to get sound and video working again.
They both failed a while back; IDK when, exactly.
On a hunch, I looked at the AV library packages in the #Debian repos. They're all at version 7.* but the older 6.* versions are still available.
On a hunch, I reinstalled the older libraries and, hey presto! Sound was working. :-) Weirdly, although videos played, there was no picture, only sound.