I have further evidence that the flex cable and/or connections between the internal speaker assembly and the logic board are indeed the cause of many if not all of the problem.
Rather than replace the cable I used the tried & true method of simply reseating it on both ends. This is not for the faint-hearted, but about as easy as it gets if you are willing to open up your Air (look at the helpful pictures on ifixit.com first). I didn't bother removing the hard disk even though I could not completely unclip the audio cable with the disk in place, but I was able to wiggle it more or less free and then reseat it firmly.
Note that I found that a lot of the screws holding the back cover and the battery in place were loose so I made sure to use a dab of lock-tite on each one. I'm wondering if the loose screws contributed to the unseating of the audio cable.
I noted that there was a lot of debate as to whether this is a hardware or software problem, and I wanted to point out that it's quite possible that it is both and that there may be more than one underlying cause. But for those of you who, like me, had their MBA sound start failing with increasing frequency until it finally stopped working altogether, I would say you almost definitely have a hardware problem. Software can fail intermittently, but it usually doesn't get worse over time.
I'm guessing that if the cable or cable connections are flaky, the logic board is losing communication with the audio assembly and causing the audio software to fail, flash video to stop working, etc. I suspect that once this fault happens, it is unable to recover until the next restart.
