My friend Jeremy Cole (https://blog.jcole.us/) who is a very qualified electronics hobbyist (and who does not live locally, so we did this remotely via pictures) has a hypothesis about my board problems.
We probed the bad boards and found that they were just shorted VCC - GND. Not a low resistance or anything -- just a plain short. Yikes.
After much diagnosis, he thinks the soldermask on my boards is misaligned, which caused the first dot of solder to short VCC to the ground plane. Here is a zoomed-in view of one of the bad boards. You can see the offset soldermask in the vias, and the suspicious area in capacitor C2 that he called out:
It is not a big stretch to imagine that this is the source of my "spikes" due to vibration on my otherwise "good" board too -- if something is really, really close, then it could indeed have an intermittent short.
I am very grateful for my friend's help, without which I would have wallowed for many weeks or months without resolution. I'm taking a day off the Airball train to hang out with my family but will resume work tomorrow and post more updates as they come in.