Mystery solved.
I probably shouldn't admit this, but it was a bad battery. The battery was the first thing I tested. Although it held voltage at rest, it would short to ground under load. I didn't have an easy way to test it under load and moved on. It eventually recovers - which is why I would get weird success with the voltage. It was pure coincidence that I figured it out, and of course after I have the bike in pieces. I had it hooked up to a battery tender while I tested connections. I noticed I was getting power back every couple minutes. Then I saw the battery tender had turned itself off (although it never indicated a bad battery).
I'm a moron, but at least I have my sanity back.