Joe, a quick update. I deleted my other post where I mentioned Media Composer (MC).
It turns out that the audio was okay in MC. Sorry about that misfire (meters in MC are a bit different). I thought I heard some noise but I listened again after a break and the playback was clean. I've been having some issues with the latest version of MC.
Looking at the audio files in Mathematica didn't yield any suggestion of bad samples, etc.
So, yes, it seems to be isolated to FCP. Thanks for mentioning your test process.
I selected the problematic.mov and first.mp4 files in FCP and exported Computer, Audio and Video, Faster Encode H.264 for both of them.
I then opened both in Izotope RX Advanced (yup, you can open most video files --- the audio is imported). Here are some instructive screenshots. The spectrogram plot makes it easy to see where the "glitches" are:
first.mp4 (exported from FCP):
problematic.mov (exported from FCP):
That was a good thought about checking metadata, instead of just focusing on the data. It's a good sanity check to be able to point to the spots where the noises are audible and see it, as well. Hope this helps.