UPDATE!
I have some info that may be leading to a small breakthrough.
First of all, even though I didn't start this thread (thanks to jennhawaiifor that), I want to thank everyone here for keeping it pretty civil. We're all justified in making this a rant fest against Apple, but that's probably not going to help fix this FaceTime problem. Also, I'm sure most, if not all of us here love Apple products and we want them to keep getting better. (Although I'd be much happier if the products weren't manufactured in Chinese sweat shops, but that's a subject for a different thread...)
So, it seems I'm the only one here so far lucky enough to be in regular contact with a senior tech, who is in regular contact with the engineers. I had sent him data awhile back that I captured with the Capture Data app that he sent me. He sent my data to the engineers and got a report back from them that it showed that while I was using FaceTime, I switched from the built in mic on my MacBook Pro, to iPhone earbuds with the mic. Well of course I didn't actually do that. I do have iPhone earbuds with the mic, but they have never been plugged into my MacBook. I had nothing plugged into my MB other than the power cord. The data was showing erroneous info about a mic that didn't exist, which would explain why the sound input cut out at the moment it thought I had switched mics. Now, why FaceTime thought this happened, of course, is still a mystery, but hopefully this is at least a starting point towards a diagnosis.
I thought that since that info is connected with my case, I would give the case number here, and maybe you can all reference it when you contact Apple. The case number is 266130739, and my name is Jim T. They probably won't need my last name, that should be enough to confirm that it's my case. I don't know if this will help but it's worth a try.