After a lot of chasing my tail, @tkimnguyen made me think to look in dmesg, where it was clear that python was crashing (mostly due to segfaults). Further research led me to find my problems were a symptom of a buggy system-installed version of Python 2.7.11 from Ubuntu 16.04:
I had installed my Plone instance last July and never looked back, dealing with this annoying bug while getting my sites up and running. It's only now that I'm getting close to launch that this became worthy of tracking down.
In hindsight, I remember finding that issue and thinking that my regular upgrading of Ubuntu would eventually fix it. Since the installer copies the system python into the plone tree, the system upgrade... it did nothing. Meanwhile, I forgot about the root cause, leading me back here today. I had the joy of realizing (and not for the first time) that the feeling I'm running around in circles isn't always just a feeling....
Thus, my solution was simply to use the installer to create a new installation, copy over my data, and run buildout in the new tree. There were even instructions to help me do that in the Plone documentation: http://docs.plone.org/manage/deploying/copy.html
I hope this helps someone save some time in the future.